Simulating large cat qubits using a shifted fock basis

ABSTRACT

A method of simulating quantum gates includes shifting a Fock basis for the simulation such that the simulation can be performed in a smaller (e.g. truncated) Hilbert dimension space. To shift the Fock basis, non-orthonormalized basis states are first defined. The defined basis states are then orthonormalized to construct orthonormalized shifted Fock basis state. Matrix elements are determined for an operator in the orthonormalized shifted Fock basis and the operator is used to simulate the quantum gate in the shifted Fock basis.

BACKGROUND

Quantum computing utilizes the laws of quantum physics to processinformation. Quantum physics is a theory that describes the behavior ofreality at the fundamental level. It is currently the only physicaltheory that is capable of consistently predicting the behavior ofmicroscopic quantum objects like photons, molecules, atoms, andelectrons.

A quantum computer is a device that utilizes quantum mechanics to allowone to write, store, process and read out information encoded in quantumstates, e.g. the states of quantum objects. A quantum object is aphysical object that behaves according to the laws of quantum physics.The state of a physical object is a description of the object at a giventime.

In quantum mechanics, the state of a two-level quantum system, orsimply, a qubit, is a list of two complex numbers whose squares sum upto one. Each of the two numbers is called an amplitude, orquasi-probability. The square of an amplitude gives a potentiallynegative probability. Hence, each of the two numbers correspond to thesquare root that event zero and event one will happen, respectively. Afundamental and counterintuitive difference between a probabilistic bit(e.g. a traditional zero or one bit) and the qubit is that aprobabilistic bit represents a lack of information about a two-levelclassical system, while a qubit contains maximal information about atwo-level quantum system.

Quantum computers are based on such quantum bits (qubits), which mayexperience the phenomena of “superposition” and “entanglement.”Superposition allows a quantum system to be in multiple states at thesame time. For example, whereas a classical computer is based on bitsthat are either zero or one, a qubit may be both zero and one at thesame time, with different probabilities assigned to zero and one.Entanglement is a strong correlation between quantum particles, suchthat the quantum particles are inextricably linked in unison even ifseparated by great distances.

A quantum algorithm is a reversible transformation acting on qubits in adesired and controlled way, followed by a measurement on one or multiplequbits. For example, if a system has two qubits, a transformation maymodify four numbers; with three qubits this becomes eight numbers, andso on. As such, a quantum algorithm acts on a list of numbersexponentially large as dictated by the number of qubits. To implement atransform, the transform may be decomposed into small operations actingon a single qubit, or a set of qubits, as an example. Such smalloperations may be called quantum gates and the arrangement of the gatesto implement a transformation may form a quantum circuit.

There are different types of qubits that may be used in quantumcomputers, each having different advantages and disadvantages. Forexample, some quantum computers may include qubits built fromsuperconductors, trapped ions, semiconductors, photonics, etc. Each mayexperience different levels of interference, errors and decoherence.Also, some may be more useful for generating particular types of quantumcircuits or quantum algorithms, while others may be more useful forgenerating other types of quantum circuits or quantum algorithms. Also,costs, run-times, error rates, error rates, availability, etc. may varyacross quantum computing technologies.

For some types of quantum computations, such as fault tolerantcomputation of large scale quantum algorithms, overhead costs forperforming such quantum computations may be high. For example for typesof quantum gates that are not naturally fault tolerant, the quantumgates may be encoded in error correcting code. However this may add tothe overhead number of qubits required to implement the large scalequantum algorithms. Also, performing successive quantum gates,measurement of quantum circuits, etc. may introduce probabilities oferrors in the quantum circuits and/or measured results of the quantumcircuits.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A illustrates a system comprising a nano-mechanical linearresonator and an asymmetrically-threaded superconducting quantuminterference device (ATS) that is configured to implement hybridacoustic-electrical qubits, according to some embodiments.

FIG. 1B illustrates a modelling of a storage mode (a) and a dump mode(b) of a hybrid acoustic-electrical qubit, wherein for large energydecay rates (K_(b)) that are significantly larger than a two-phononcoupling rate (g₂) the dump mode can be adiabatically eliminated, suchthat the hybrid acoustic-electrical qubit can be modeled as having asingle phonon decay rate (K₁) and being driven by a two phonon drivehaving a two-phonon decay rate (K₂), according to some embodiments.

FIG. 2 illustrates a Foster network representing a one dimensionalphononic-crystal-defect resonator (PCDR), according to some embodiments.

FIG. 3 illustrates a system comprising a plurality of nano-mechanicallinear resonators and an asymmetrically-threaded superconducting quantuminterference device (ATS) that is configured to provide multi-modestabilization to hybrid acoustic-electrical qubits implemented via theplurality of nano-mechanical linear resonators, according to someembodiments.

FIG. 4 illustrates a system comprising a plurality of nano-mechanicallinear resonators and an asymmetrically-threaded superconducting quantuminterference device (ATS) that is configured to provide multi-modestabilization to hybrid acoustic-electrical qubits implemented via theplurality of nano-mechanical linear resonators, wherein a microwavefilter suppresses correlated decay processes, according to someembodiments.

FIG. 5 illustrates a process of stabilizing a nano-mechanical resonatorusing an asymmetrically-threaded superconducting quantum interferencedevice (ATS), according to some embodiments.

FIG. 6 illustrates a process of stabilizing multiple nano-mechanicalresonators using a multiplexed ATS, according to some embodiments.

FIG. 7 illustrates a data error occurring when measuring input errorsfor a set of qubits, wherein the data error causes multiple distinctsyndromes, according to some embodiments.

FIG. 8 illustrates a measurement of logical Z for a repetition code anda corresponding circuit for measuring the logical Z for the repetitioncode, according to some embodiments.

FIG. 9 illustrates a circuit for preparing Q=SHS, wherein the CNOT gateis a single physical CNOT and Y is applied if the measurement outcome is−1, according to some embodiments.

FIG. 10 illustrates a circuit for preparing S, wherein the CNOT gate isa single physical CNOT and Z is applied if the measurement outcome is−1, according to some embodiments.

FIG. 11A illustrates a circuit for implementing a logical Toffoli gateusing Toffoli magic state injection, wherein X and Z basis are measured,according to some embodiments.

FIG. 11B illustrates a table of Clifford error corrections to be appliedbased on the Z and X measurement basis of the circuit shown in FIG. 11A,according to some embodiments.

FIG. 12 illustrates a circuit for implementing a logical CZ gate usingtransversal CNOT gates and S gates, according to some embodiments.

FIG. 13 illustrates a circuit for preparing the computational basisstate |ψ₁

, according to some embodiments.

FIG. 14 illustrates a circuit for implementing a first step of a Toffolimagic state preparation using a controlled g_(A) gate, wherein errorcorrection (EC) is performed for one or more rounds using a STOPalgorithm, according to some embodiments.

FIG. 15 illustrates circuits for implementing a second step of theToffoli magic state preparation, wherein the measurement of g_(A) isrepeated a number of times corresponding to a code distance (d) minusone divided by two, wherein a round of repetition code stabilizermeasurements are performed between rounds of measuring g_(A), andwherein the protocol is aborted and started anew if any of the errordetection measurements or g_(A) measurements are non-trivial, accordingto some embodiments.

FIG. 16 illustrates growing the computational basis state of |ψ₁

, from a first code distance (d₁) to a second code distance (d₂),according to some embodiments.

FIG. 17 illustrates a circuit for measuring g_(A) for a computationalbasis state |ψ₁

with a code distance of three, according to some embodiments.

FIG. 18 illustrates an alternative circuit for measuring g_(A) for acomputational basis state |ψ₁

using a flag qubit, according to some embodiments.

FIG. 19A illustrates an implementation of the g_(A) measurement for adistance 5 repetition code prepared using a GHZ state, according to someembodiments.

FIG. 19B illustrates a circuit equivalent for implementing the g_(A)measurement shown in FIG. 19A, according to some embodiments.

FIG. 20A illustrates high-level steps of a protocol for implementing aSTOP algorithm, according to some embodiments.

FIG. 20B illustrates high-level steps for determining a parameter (miff)used in the STOP algorithm, according to some embodiments.

FIG. 21 illustrates high-level steps of a protocol for growing arepetition code from a first code distance to a second code distanceusing a STOP algorithm, according to some embodiments.

FIG. 22 illustrates high-level steps of a protocol for implementing alogical Toffoli gate using a bottom-up approach with Toffoli magic stateinjection, according to some embodiments.

FIG. 23 illustrates high-level steps for distilling a low-error ratelogical Toffoli gate using multiple ones of the Toffoli magic statesprepared using a bottom-up approach as described in FIG. 22, accordingto some embodiments.

FIG. 24 illustrates a layout of multiple bottom up Toffoli magic statesthat are used to distill low-error rate logical Toffoli gates, accordingto some embodiments.

FIG. 25 illustrates a gadget for injection of CCZ gates using a |CCZ

magic state and a gadget for generalized CCZ injection for a unitary,according to some embodiments.

FIG. 26 illustrates a circuit for implementing distillation of twolow-error rate logical Toffoli gates (CCZ gates) from eight magic stateinputs, according to some embodiments.

FIG. 27 illustrates a Litinski diagram for performing lattice surgeryrealization of a distillation of eight Toffoli magic states to yield twolow-error rate logical Toffoli gates, according to some embodiments.

FIG. 28 illustrates a process for distilling low-error rate logicalToffoli gates from a plurality of noisy Toffoli magic states, accordingto some embodiments.

FIG. 29A illustrates a process of distilling two low-error rate logicalToffoli gates from eight noisy Toffoli magic states, according to someembodiments.

FIG. 29B illustrates a process of distilling a low-error rate logicalToffoli gate from two noisy Toffoli magic states, according to someembodiments.

FIG. 30 illustrates an example method of performing lattice surgery todistill a low-error rate logical Toffoli gate from a plurality of noisyToffoli magic states, according to some embodiments.

FIG. 31 illustrates a circuit for performing measurements of a readoutqubit for a set of error correction gates in parallel with performing anext round of error correction gates, according to some embodiments.

FIG. 32 illustrates a more detailed circuit for performing measurementsof a readout qubit for a set of error correction gates in parallel withperforming a next round of error correction gates, according to someembodiments.

FIG. 33 illustrates the more detailed circuit for performingmeasurements of a readout qubit for a set of error correction gates inparallel with performing a next round of error correction gates, whereinthe circuit includes a deflation of the ancilla qubit prior to a swap tothe readout qubit and wherein the measurement comprises a paritymeasurement of the readout qubit, according to some embodiments.

FIG. 34 is a process flow diagram illustrating using a switch operatorto excite a readout qubit such that a subsequent round of errorcorrection gates can be applied in parallel with performing measurementsof the readout qubit, according to some embodiments.

FIG. 35 is a process flow diagram illustrating a process for usingdeflation to perform measurements of a qubit, according to someembodiments.

FIG. 36A is a process flow diagram illustrating a process for deflatinga cat qubit and measuring a “b” mode of the deflated cat qubit todetermine information about a first mode of the deflated cat qubit,according to some embodiments.

FIG. 36B is a process flow diagram illustrating another process fordeflating a cat qubit and measuring a “b” mode of the deflated cat qubitto determine information about a first mode of the deflated cat qubit,according to some embodiments.

FIG. 37 is a process flow diagram illustrating a process for evolving acat qubit via three wave or higher mixing Hamiltonian and performing ahomodyne, heterodyne, or photo detection of the evolved cat qubit tomeasure a measured property of another bosonic mode of the cat qubit,according to some embodiments.

FIG. 38 is a process flow diagram illustrating a process of utilizing ashifted Fock basis to simulate a cat qubit with (|α|²>>1), according tosome embodiments.

FIG. 39 is a block diagram illustrating an example computing device thatmay be used in at least some embodiments.

While embodiments are described herein by way of example for severalembodiments and illustrative drawings, those skilled in the art willrecognize that embodiments are not limited to the embodiments ordrawings described. It should be understood, that the drawings anddetailed description thereto are not intended to limit embodiments tothe particular form disclosed, but on the contrary, the intention is tocover all modifications, equivalents and alternatives falling within thespirit and scope as defined by the appended claims. The headings usedherein are for organizational purposes only and are not meant to be usedto limit the scope of the description or the claims. As used throughoutthis application, the word “may” is used in a permissive sense (i.e.,meaning having the potential to), rather than the mandatory sense (i.e.,meaning must). Similarly, the words “include,” “including,” and“includes” mean including, but not limited to. When used in the claims,the term “or” is used as an inclusive or and not as an exclusive or. Forexample, the phrase “at least one of x, y, or z” means any one of x, y,and z, as well as any combination thereof.

DETAILED DESCRIPTION

The present disclosure relates to methods and apparatus for implementinga universal gate set for quantum algorithms that are fault-tolerant andthat efficiently use resources.

In many circumstances, the overhead cost of performing universalfault-tolerant quantum computation for quantum algorithms may be high.To perform such fault-tolerant quantum computations, magic statedistillation is often used. For example, magic state distillation may beused for simulating non-Clifford gates in a fault tolerant way. However,since magic state distillation circuits are not fault-tolerant, theClifford operations must be encoded in a large distance code in order tohave comparable failure rates with the magic states being distilled.

In order to perform quantum computations, universal fault-tolerantquantum computers may be required to be built with the capability ofimplementing all gates from a universal gate set with low logical errorrates. Further, the overhead cost for achieving such low error rates mayneed to be low. Transversal gates are a natural way to implement suchfault-tolerant gates. However, as is known from the Eastin-Knilltheorem, given any stabilizer code, there will always be at least onegate in a universal gate set that cannot be implemented usingtransversal operations at the logical level.

In order to deal with this issue, several fault-tolerant methods forimplementing gates in a universal gate set have been explored. However,magic state distillation remains a leading candidate in theimplementation of a universal fault-tolerant quantum computer. However,the costs of performing magic state distillation remains high. One ofthe reasons for the high costs of magic state distillation is that theClifford circuits used to distill the magic states are often notfault-tolerant. Consequently, the Clifford gates are encoded in someerror correcting code (often the surface code) to ensure that thesegates have negligible error rates compared to the magic states beinginjected.

In some embodiments, efficiently implementing a universal gate set mayinvolve multiple layers of a quantum computer/quantum algorithm. Forexample at a lowest layer, building blocks of a quantum computer mayinclude nano-mechanical resonators that are controlled using anasymmetrically-threaded superconducting quantum interference device(asymmetrically-threaded SQUID or ATS). The nano-mechanical resonatorsmay be configured to resonate at one or more frequencies and may becoupled to the ATS, wherein the ATS controls the phononic modes. Alsothe ATS may be coupled to a bandpass filter and then an opentransmission line that enables photons from the ATS to be adsorbed bythe environment. At a next level, error correction may be implementedfor the quantum computer comprising nano-mechanical resonators and anATS. For example error corrected codes may be built that utilize the ATSand phononic modes of the nano-mechanical resonators to detect and/orcorrect errors. At yet another level, gates may be implemented for thequantum computer using the error corrected codes as inputs or outputs tothe gates. Also, qubits of the gates may be error corrected. At yet ahigher level logical gates may be built that utilize one or more of thephysical gates. Note that while several of the protocols describedherein, such as the STOP algorithm, bottom-up approach to preparingToffoli gates, the top-down distillation of Toffoli gates, measurementtechniques, and/or shifted Fock basis simulations are described in termsof a system that utilizes nano-mechanical resonators that implementhybrid acoustic-electrical qubits, in some embodiments other hardwaretypes may be used, such as those that implement electromagnetic qubits.

Asymmetrically Threaded Superconducting Quantum Interference Device(ATS)-Phononic Hybrid System

In some embodiments, a circuit for use in a quantum computer maycomprise nano-mechanical linear resonators and an asymmetricallythreaded superconducting quantum interference device (SQUID, ATS). Thenano-mechanical resonators and ATS may implement qubits that are hybridacoustic-electrical qubits, for example as opposed to electromagneticqubits. In some embodiments, both the nano-mechanical resonators and ATSmay be situated on a same component and may provide for easily extendinga system to include additional components with additionalnano-mechanical resonators that implement additional hybridacoustic-electrical qubits. This may also enable scaling of a number ofqubits needed for a quantum computer by including more or fewercomponents. Such an approach may allow for simpler extension and scalingthan a system wherein components that implement qubits are integratedinto a single chip, and newly designed chips are required to extend orscale the system to have more or fewer qubits. As used herein, the terms“mechanical”. “acoustic”, “phononic”, etc. may be used to describemechanical circuits as opposed to electromagnetic circuits.

In some embodiments, more phononic resonators (e.g. nano-mechanicalresonators) may be connected to a same control circuit, such as an ATS,than is possible for electromagnetic resonators. This is due, at leastin part, to the smaller size of the phononic resonators as compared toelectromagnetic resonators. However, in such systems cross-talk betweenthe phononic resonators coupled to the same control circuit must beaddressed in order to avoid errors. Multiplexed control of phononicresonators using a common control circuit, such as an ATS, is furtherdiscussed in detail below.

In some embodiments, a structure of a chip comprising phononicresonators may take the form of a planar circuit with metal componentsthat form superconducting circuits, such as the ATS. The ATS may bephysically connected via wire leads to very small (e.g. micron-sized ornano-sized) suspended mechanical devices, such a linear nano-mechanicalresonator. The suspended mechanical devices may be located on a samechip with the ATS circuit or may by located on a separate chip that hasbeen heterogeneously integrated via a flip chip, or similar component,with a bottom chip comprising the ATS and/or additional suspendedmechanical devices, e.g. other mechanical resonators.

In some embodiments, electrical connections to the ATS may be laid ontop of a piezoelectric material that has been etched into a pattern toform the nano-mechanical resonators. In some embodiments, differentvariables, such as piezoelectric coefficient, density, etc. may affecthow strongly coupled the ATS is to the mechanical resonators. Thiscoupling may be expressed in terms of a phonon coupling rate in theHamiltonian for the system.

When coupling a nano-structure, such as a nano-mechanical resonator, toan electrical circuit, very small capacitors are required since thenano-structure components, e.g. nano-mechanical resonators, are alsovery small. Typically in an electrical circuit, such as an ATS circuit,there are other capacitances. Since the capacitor for the nano-structureis very small, these other capacitances in the circuit may lower thesignal voltage and thus dilute a signal directed to one of thenano-components, such as a nano-mechanical resonator. However, to dealwith this issue, a high-impedance inductor may be coupled in the controlcircuit between the ATS and the nano-mechanical resonator. Thehigh-impedance inductor may have a very low parasitic capacitance, thuselectrical fields directed at the nano-mechanical resonators may act onthe nano-mechanical resonators with only minimal dilution due tocapacitance of the inductor (e.g. parasitic capacitance). Also, the highimpedance inductor may suppress loss mechanisms.

In some embodiments, the non-linear coupling of the nano-mechanicalresonators may be given by g₂â²{circumflex over (b)}^(†)+h.c., where g₂is a coupling rate between a storage mode (a) and a dump mode (b). Insome embodiments, the non-linearity may be implemented using anasymmetrically threaded SQUID (superconducting quantum interferencedevice), also referred to herein as an “ATS.” The ATS may comprise asuperconducting quantum interference device (SQUID) that has been splitapproximately in the middle by a linear inductor. In its most generalform, the ATS potential is given by the following equation:

${U( \hat{\phi} )} = {{\frac{1}{2}E_{L,b}{\hat{\phi}}^{2}} - {2E_{j}{\cos( \phi_{\Sigma} )}{\cos( {\hat{\phi} + \phi_{\Delta}} )}} + {2\Delta\; E_{j}{\sin( \phi_{\Sigma} )}{\sin( {\hat{\phi} + \phi_{\Delta}} )}}}$

In the above equation, {circumflex over (ϕ)} is the phase differenceacross the ATS, ϕ_(Σ)

(ϕ_(ext,1)+ϕ_(ext,2))/2, ϕ_(Δ)

(ϕ_(ext,1)−ϕ_(ext,2))/2, and ϕ_(ext,1)(ϕ_(ext,2)) is the magnetic fluxthreading the left (right) loop, in units of the reduced magnetic fluxquantum Φ₀ ²=h/2E. Here

${E_{L,b} = {\Phi_{0}^{2}/L_{b}}};{E_{j} = {( {E_{j,1} + E_{j,2}} )/2}};{{{and}\mspace{14mu}\Delta\; E_{j}} = \frac{( {E_{j,1} - E_{j,2}} )}{2}}$is the junction asymmetry. This ATS potential can be further simplifiedby tuning ϕ_(Σ) and ϕ_(Δ) with two separate flux lines. For example,FIG. 1A illustrates ATS 102 included in control circuit 100, wherein ATS102 includes separate flux lines 108 and 110. Note that FIG. 1A includesATS 102 in control circuit 100 and also an enlarged depiction of ATS 102adjacent to control circuit 102 that shows ATS 102 in more detail. Theflux lines may be set such that:

$\phi_{\Sigma} = {\frac{\pi}{2} + {{\epsilon_{p}(t)}\mspace{14mu}{and}}}$$\phi_{\Delta} = \frac{\pi}{2}$

In the above equations, ∈_(p)(t)=∈_(p,0) cos(ω_(p)t) is a smallalternating current (AC) component added on top of the direct current(DC) basis. At this bias point, and assuming that |∈_(p)|<<1 then theequation above for U({circumflex over (ϕ)}) can be reduced to:

${U(\phi)} = {{\frac{1}{2}E_{L,b}{\overset{\hat{}}{\phi}}^{2}} - {2E_{j}{\epsilon_{p}(t)}{\sin( \overset{\hat{}}{\phi} )}} + {2\Delta E_{j}{{\cos( \overset{\hat{}}{\phi} )}.}}}$

Using the control circuit 100 shown in FIG. 1A, quantum information maybe stored in a state of a linear mechanical resonator. For examplequantum information may be stored in storage mode 106. The storedquantum information may also be autonomously error corrected by way ofartificially induced two-phonon driving and two-phonon decay controlledby the ATS. These two phonon processes are induced through thenon-linear interaction g₂ â²{circumflex over (b)}^(†)+h·c. between thestorage mode a and an ancillary mode b, called the dump, such as dumpmode 104 shown in FIG. 1A. The dump mode is designed to have a largeenergy decay rate K_(d) so that it rapidly and irreversibly “dumps” thephotons it contains into the environment. If K_(d) is much larger (e.g.˜10× or more) than the coupling rate g₂, then the dump mode can beadiabatically eliminated from the Hamiltonian, for example as shown inFIG. 1B. For example, as shown on the right side of FIG. 1B, theemission of phonon pairs via g₂ â²{circumflex over (b)}^(†) can beaccurately modeled as a dissipative process described by a dissipator˜D[a²]. Additionally, if the dump mode is linearly driven as ∈*be^(−ω)^(d) ^(t)+h·c. this provides the required energy to stimulate thereverse process g*₂(a⁺²)b, which in the adiabatic elimination, as shownin FIG. 1B, can be modeled as an effective two-phonon drive. Altogether,the dynamics can be accurately modeled through the equation:

${\frac{d\rho}{dt} = {K_{2}{D\lbrack {a^{2} - \alpha^{2}} \rbrack}}},{{{where}\mspace{14mu}\alpha} = {{{\epsilon/g_{2}}\mspace{14mu}{and}\mspace{14mu} k_{2}} = {4{g_{2}^{2}/K_{d}}}}}$

The steady states of the dynamics of the system shown in FIG. 1B are thecoherent states |α>, |−α>, or any arbitrary superposition of the two.This protected subspace can be used to encode a qubit through thefollowing definition of a logical basis: |0_(L)

=|α

, |1_(L)

=|−α

. Qubits encoded in this way are effectively protected from X errors(e.g. bit flips) because the bit-flip rate decays exponentially with thecode distance |α|², as long as K₂|α|²>>K₁, wherein K₁ is the ordinary(e.g. single-photon) decay rate of the storage mode. Since |α|²˜1, thiscondition is generally equivalent to K₂/K₁>>1. However, Z errors (e.g.phase flips) may not be protected by this code.

As discussed above, an ATS is formed by splitting a SQUID with a linearinductor. The magnetic flux threading of each of the two resulting loopsof the ATS can be controlled via two nearby on-chip flux lines, such asflux lines 108 and 110 shown in FIG. 1A. These flux lines can be tunedto appropriate values and can send radio frequency (rf) signals atappropriate frequencies for a desired non-linear interaction to beresonantly activated in the nano-mechanical resonator. The dump mode104, may further be strongly coupled to a dump line of characteristicimpedance Z₀, which induces a large energy decay rate as required.

In some embodiments, the nano-mechanical storage resonator (e.g. storage106) may be a piezoelectric nano-mechanical resonator that supportsresonances in the GHz range. These resonances may be coupled tosuperconducting circuits of the control circuit 100 via smallsuperconducting electrodes (e.g. terminals) that either directly touchor closely approach the vibrating piezoelectric region of thenano-mechanical resonators. The values of the nonlinear coupling rateg₂, the two-phonon dissipation rate k₂, and the ratio K₂/K₁ can becalculated as follows:

First, compute the admittance Y_(m)(ω) seen at the terminals of thenano-mechanical resonator using a finite element model solver. Next,find an equivalent circuit using a Foster synthesis algorithm (furtherdiscussed below). Then, diagonalize the combined circuit and compute thezero-point phase fluctuations ϕ_(a,zp) and ϕ_(b,zp) Furthermore, computethe dissipation rates k_(b) and k₁ of the eigenmodes. Next compute

$g_{2} = {( \frac{E_{j}}{h} )\epsilon_{0}\phi_{a,{zp}}^{2}{\phi_{b,{zp}}^{2}/{2.}}}$Also, compute k₂=4g₂ ²/k_(d).

In some embodiments, a nano-mechanical element, such as thenano-mechanical resonator that implements storage mode 106 and dump mode104 may be represented as an equivalent circuit that accurately capturesits linear response. This can be done using Foster synthesis if theadmittance Y_(m)(ω) seen from the terminals of the mechanical resonatoris known. For example, the admittance may be computed using finiteelement modeling. In some embodiments, a Foster network may be used toaccurately represent a one-dimensional (e.g. linear)phononic-crystal-defect resonator (PCDR), which may be a type ofnano-mechanical resonator used in some embodiments. In some embodiments,the dump resonator may be modeled as having a fixed impedance, such as 1kilo ohms.

For example FIG. 2 illustrates a version of control circuit 100 that hasbeen represented using a Foster network (e.g. equivalent circuit 200).In its simplest form, equivalent circuit 200 may be represented as ‘a DCcapacitance’ in series with an LC block (e.g. L represents and inductorand C represents a capacitor for the LC block), wherein an additionalresistor is inserted to include the effects of the loss in theresonator. For example, Foster network 210 is modeled to includecapacitor 204, inductor 206, and resistor 208. The linear part of thedump resonator (including the inductor that splits the ATS) can also berepresented as an LC block, such as LC block 212. In this representationthe dump resonator (e.g. 212) and the storage resonator (e.g. 210) arerepresented as two linear circuits with a linear coupling and cantherefore be diagnolized by a simple transformation of coordinates. Forexample, FIG. 2 illustrates a diagnolized circuit representation 214.The resulting “storage-like” ({circumflex over (α)}) and “dump-like”({circumflex over (b)}) eigenmodes both contribute to the total phasedrop across the ATS. For example, {circumflex over (ϕ)}=φ_(a),(â+â^(†))+φ_(b) ({circumflex over (b)}+{circumflex over (b)}^(†)). Thesemodes therefore mix the via the ATS potential, which may be redefined asU({circumflex over (ϕ)})→U({circumflex over (ϕ)})—E_(L,b){circumflexover (φ)}²/2 because the inductor has already been absorbed into thelinear network. The zero-point phase fluctuations of each mode are givenby:

$\varphi_{k,j} = {\sqrt{\frac{h}{2\omega_{k}}}( {C^{{- 1}/2}U} )_{jk}}$

In the above equation C is the Maxwell capacitance matrix of thecircuit. U is the orthogonal matrix that diagnolizesC^(−1/2)L⁻¹C^(−1/2), where L⁻¹ is the inverse inductance matrix. Theindex k∈{a, b} labels the mode and j labels the node in question. Notethat in some instances as described herein the notation of j may beomitted because it is clear from context, e.g. the node of interest isthe one right above the ATS.

The way in which the ATS mixes the modes is explicit given thethird-order term in the Taylor series expansion of the sin({circumflexover (ϕ)}) contains terms of the form â²{circumflex over (b)}†+h·c.,which is the required coupling. This is a reason for using the ATS asopposed to an ordinary junction, which has a potential ˜cos({circumflexover (ϕ)}).

For analysis the pump and drive frequencies may be set toω_(p)=2ω_(a)−ω_(b) and ω_(d)=ω_(b). This brings the termsg₂â²{circumflex over (b)}^(†)+h.c. into resonance allows the other termsin the rotating wave approximation (RWA) to be dropped. The coupling isgiven by g₂=∈₀E_(j)φ_(a) ²φ_(b)/2 h. Additionally, a linear drive∈*_(d){circumflex over (b)}+h.c. at frequency ω_(d)=ω_(b) is added tosupply the required energy for the two-photon drive.

Multi-Mode Stabilization/ATS Multiplexing

In some embodiments, the scheme as described above may be extended to beused in a multi-mode setting, in which N>1 storage resonators aresimultaneously coupled to a single dump+ATS. This may allow for the catsubspaces of each of the storage modes to be stabilized individually.For example, a dissipator of the form Σ_(n)D[a_(n) ²−α²]. However, inorder to avoid simultaneous or coherent loss of phonons from differentmodes (which fails to stabilize the desired subspaces), an incoherentdissipator is required. This can be achieved if the stabilization pumpsand the drives for the different modes are purposefully detuned asfollows:H=Σ _(m)(∈*_(m) ^((d))(t)b ^(†) +h.c.)+Σ_(m,i,j)(g* _(ij) ^((m))(t)a_(i) a _(j) b ^(†) +h.c.), where∈*_(m) ^((d))(t)=∈*_(m) ^((d)) e ^(iΔ) ^(m) ^(t) and g* _(ij)^((m))(t)=g* ₂ e ^(i(2ω) ^(m) ^(−ω) ^(i) ^(+Δ) ^(m) ^()t)

In the above equation ω_(m) ^((p))=2ω_(m)−ω_(b)+Δ_(m) and ω_(m)^((d))=ω_(b)−Δ_(m) are the pump and drive frequencies for mode m. Bydetuning the pumps, the pump operators of different modes can rotatewith respect to each other. If the rotation rate is larger than k₂ thenthe coherences of the form a_(i) ²ρ(a_(j) ^(†))² in the Lindbladianvanish in a time averaged sense. The drive de-tunings allow the pumpsand drives to remain synchronized even though the pumps have beendetuned relative to one another.

In some embodiments, the modes a₁ and a₂ may be simultaneouslystabilized using a multiplexed ATS, wherein the pumps have been detuned.Simulations may be performed to determine the detuning parameters usingthe simulated master equation, as an example:

$\overset{.}{\rho} = {{- {i\lbrack {{{\frac{\Delta}{2}a_{1}^{\dagger}a_{1}} + \ ( {{\epsilon_{2}e^{i\Delta t}a_{1}^{\dagger 2}} + {\epsilon_{2}a_{2}^{\dagger 2}} + {h.c.}} )},\rho} \rbrack}} + {k_{2}{D\lbrack {a_{1}^{2} + a_{2}^{2}} \rbrack}(\rho)}}$Bandwidth Limits

The above described tuning works best when the detuning Δ is relativelysmall as compared to k_(b). This is due to the fact that, unlike thesingle-mode case, where k₂=4 g₂ ²/k_(b), the two-phonon decay of themulti-mode system is given by:

$k_{2,n} = {\frac{4{g}^{2}}{k_{b}}\frac{1}{1 + {4( {\Delta_{n}/k_{b}} )^{2}}}}$

The Lorentzian suppression factor can be understood by the fact thatphotons/phonons emitted by the dump mode as a result of stabilizing moden are emitted at a frequency ω_(b)+Δ_(n) and are therefore “filtered” bythe Lorentzian line-shape of the dump mode which has linewidth k_(b).This sets an upper bound on the size of the frequency region that thede-tunings are allowed to occupy. Furthermore, in some embodiments, thede-tunings Δ_(n) may all be different from each other by amount greaterthan k₂ in order for the dissipation to be incoherent. In a frequencydomain picture, the spectral lines associated with emission ofphotons/phonons out of the dump must all be resolved. This, also sets alower bound on the proximity of different tunings. As such, since anupper bound and lower bound are set, bandwidth limits for the de-tuningsmay be determined. Also, taking into account these limitations, an upperbound on the number of modes that can be simultaneously stabilized by asingle dump can also be determined. For example, if de-tunings areselected to be Δ_(n)=nΔ, with Δ˜k₂, then the maximum number of modesthat may be simultaneously stabilized may be limited asN_(max)˜k_(b)/Δ˜k_(b)/k₂. As a further example, for typical parameters,such as k_(b)/2π˜10 MHz and k₂/2π˜1 MHz, this results in bandwidthlimits that allow for approximately 10 modes to be simultaneouslystabilized.

For example, FIG. 3 illustrates a control circuit 300 that includes asingle dump resonator 302 that stabilizes multiple storage resonators304.

Use of a High-Impedance Inductor to Enhance Coupling Between a DumpResonator and One or More Storage Resonators

In some embodiments, the coupling rate g₂ may be increased by using ahigh impedance inductor. This is because g₂ depends strongly on theeffective impedance Z_(d) of the dump resonator. For example, g₂˜Z_(d)^(5/2). Thus, in some embodiments, using a large inductor in the ATS mayresult in a large effective impedance Z_(d). In some embodiments, theinductor chosen to be included in the ATS circuit may be sufficientlylinear to ensure stability of the dump circuit when driven stronglyduring stabilization. For example, a high impedance inductor used maycomprise a planar meander or double-spiral inductor, a spiral inductorwith air bridges, an array with a large number of (e.g. greater than 50)highly transparent Josephson junction, or other suitable high impedanceinductor.

Filtering in Multi-Mode Stabilization/Multiplexed ATS

In some embodiments, microwave filters (e.g. metamaterial waveguides)may be used to alleviate the limitations with regard to bandwidth limitsas discussed above. Such filters may also be used to eliminatecorrelated errors in multiplexed stabilization embodiments. For example,FIG. 4 illustrates control circuit 400 that includes a single dumpresonator 404, multiple storage resonators 406, and a filter 402.

More specifically, when stabilizing multiple storage modes with the samedump resonator and ATS device a number of cross-terms appear in theHamiltonian that would otherwise not be there in the single-mode case.For example, these terms take the form of g₂a_(j)a_(k)b⁺e^(−ivt). Afteradiabatic elimination of the b mode (for example as discussed in regardto FIG. 1B), these terms effectively become jump operators of the formk_(2,eff)a_(j)a_(k)e^(−ivt) Unlike the desired jump processes k₂, a_(j)², which result in the individual stabilization of the cat subspace ofeach resonator, the correlated decay terms result in simultaneous phaseflips of the resonators j and k. For example, these correlated errorscan be damaging to the next layer of error correction, such as in arepetition or striped surface code.

In some embodiments, in order to filter out the unwanted terms in thephysical Hamiltonian that give rise to effective dissipators that causecorrelated phase flips, the de-tunings of the unwanted terms may belarger than half the filter bandwidth. This may result in an exponentialsuppression of the unwanted terms. Said another way, the de-tunings andfilter may be selected such that detuning of the effective Hamiltonianis larger than half the filter bandwidth. Moreover, the filter mode(along with the dump mode) may be adiabatically eliminated from themodel in a similar manner as discussed in FIG. 1B for the adiabaticelimination of the dump mode. This may be used to determine an effectivedissipator for a circuit such as control circuit 400 that includes bothdump resonator 404 and filter 402.

As discussed above, correlated phase errors may be suppressed by afilter if the corresponding emitted photons have frequencies outside ofthe filter bandwidth. In some embodiments, all correlated phase errorsmay be simultaneously suppressed by carefully choosing the frequenciesof the storage modes. For example cost functions may be used taking intoaccount a filter bandwidth to determine optimized storage frequencies.For example, in some embodiments a single ATS/dump may be used tosuppress decoherence associated with all effective Hamiltonians for 5storage modes. In such embodiments, all dominant sources of stochastic,correlated phase errors in the cat qubits may be suppressed.

Multi-Terminal Mechanical Resonators

In some embodiments, nano-mechanical resonators, such as those shown inFIGS. 1-4 may be designed with multiple terminals that allow a givennano-mechanical resonator to be coupled with more than one ATS/controlcircuit. For example a single connection ATS may include a groundterminal and a signal terminal, wherein the signal terminal couples witha control circuit comprising an ATS. In some embodiments, amulti-terminal nano-mechanical resonator may include more than onesignal terminal that allows the nano-mechanical resonator to be coupledwith more than one control circuit/more than one ATS. For example, insome embodiments, a nano-mechanical resonator may include three or moreterminals that enable the nano-mechanical resonator to be coupled withthree or more ATSs. If not needed an extra terminal could be coupled toground, such that the multi-terminal nano-mechanical resonator functionslike a single (or fewer) connection nano-mechanical resonator. In someembodiments, different signal terminals of a same nano-mechanicalresonator may be coupled with different ATSs, wherein the ATSs may beused to implement gates between mechanical resonators, such as a CNOTgate. For example, this may allow for implementation of gates on thestabilizer function.

Example Physical Gate Implementations

Recall the Hamiltonian of a system comprising of multiple phononic modesâ_(k) coupled to a shared ATS mode {circumflex over (b)}:

$\hat{H} = {{\sum\limits_{k = 1}^{N}{\omega_{k}{\overset{\hat{}}{a}}_{k}^{\dagger}{âk}}} + {\omega_{b}{\overset{\hat{}}{b}}^{\dagger}\overset{\hat{}}{b}} - {2E_{j}{\epsilon_{p}(t)}{\sin( {{\sum\limits_{k = 1}^{N}{\overset{\hat{}}{\phi}}_{k}} + {\overset{\hat{}}{\phi}}_{b}} )}}}$

wherein {circumflex over (ϕ)}_(k)≡φ_(k)(â_(k)+â_(k) ^(†)) and{circumflex over (ϕ)}_(b)≡φ_(b)({circumflex over (b)}+{circumflex over(b)}^(†)). Also, φ_(k) and φ_(b) quantify zero-point fluctuations of themodes â_(k) and {circumflex over (b)}. To simplify the discussion,neglect small frequency shifts due to the pump ∈_(p)(t) for the momentand assume that the frequency of a mode is given by its bare frequency(in practice, however, the frequency shifts need to be taken intoaccount; see below for the frequency shift due to pump). Then, in therotating frame where every mode rotates with its own frequency, thefollowing is obtained:

${\hat{H}}_{rot} = {{- 2}E_{j}{\epsilon_{p}(t)}{\sin( {{\sum\limits_{k = 1}^{N}{\varphi_{k}{\overset{\hat{}}{a}}_{k}e^{{- \omega_{k}}t}}} + {{h.c.{+ \phi_{b}}}\overset{\hat{}}{b}e^{{- \omega_{b}}t}} + {h.c.}} )}}$where φ_(k) and φ_(b) quantify zero-point fluctuations of the modesâ_(k) and {circumflex over (b)}. Note that the rotating frame has beenused where each mode rotates with its own frequency.

First, a linear drive on a phononic mode, say â_(k), can be readilyrealized by using a pump ∈_(p)(t)=∈_(p) cos(ω_(p)t) and choosing thepump frequency ω_(p) to be the frequency of the mode that is to bedriven, that is, ω_(p)=ω_(k). Then, by taking only the leading orderlinear term in the sine potential (e.g., sin({circumflex over(x)})≈{circumflex over (x)} we get the desired linear drive:Ĥ _(rot)=−2E _(j)∈_(p)φ_(k)(â _(k)+{circumflex over (α)}_(k) ^(†))+H′where H′ comprises fast-oscillating terms such as−E_(j)∈_(p)(φ_(l)â_(l)e^(−i(ω) ^(l) ^(−ω) ^(k) ^()t)+h.c.) with l≠k andE_(j)∈_(p)(φ_(b){circumflex over (b)}e^(−i(ω) ^(b) ^(−ω) ^(k)^()t)+h.c.) as well as other terms that rotate even faster. Since thefrequency differences between different modes are on the order of 100MHz but |∈_(z)|/(2π) is typically much smaller than 100 MHz, the fasteroscillating terms can be ignored using a rotating wave approximation(RWA).

To avoid driving unwanted higher order terms, one may alternativelydrive the phononic mode directly, at the expense of increased hardwarecomplexity, instead of using the pump ∈_(p)(t) at the ATS node.

Now moving on to the implementation of the compensating Hamiltonian fora CNOT gate. For example a compensating Hamiltonian for a CNOT gate mayhave the form:

${\hat{H}}_{CNOT} = {\frac{\pi}{4\alpha T}( {{\overset{\hat{}}{a}}_{1} + {\overset{\hat{}}{a}}_{1}^{\dagger} - {2\alpha}} )( {{{\overset{\hat{}}{a}}_{2}^{\dagger}{\overset{\hat{}}{a}}_{2}} - \alpha^{2}} )}$

Without loss of generality, consider the CNOT gate between the modesâ_(i) (control) and â₂ (target). Note that Ĥ_(CNOT) comprises anoptomechanical coupling

${\frac{\pi}{4\alpha T}( {{\overset{\hat{}}{a}}_{1} + {\overset{\hat{}}{a}}_{1}^{\dagger}} ){\overset{\hat{}}{a}}_{2}^{\dagger}}{\overset{\hat{}}{a}}_{2}$between two phononic modes, a linear drive on the control mode

${{- ( \frac{\pi\alpha}{4T} )}( {{\overset{\hat{}}{a}}_{1} + {\overset{\hat{}}{a}}_{1}^{\dagger}} )},$and a selective frequency shift of the target mode

${- ( \frac{\pi}{2T} )}{\overset{\hat{}}{a}}_{2}^{\dagger}{{\overset{\hat{}}{a}}_{2}.}$To realize the optomechanical coupling, one might be tempted to directlydrive the cubic term â₁â₂ ^(†)â₂+h.c., in the sine potential via a pump∈_(p)(t)=∈_(p) cos(ω_(p)t). However, the direct driving scheme is notsuitable for a couple of reasons: since the term â₁â₂ ^(†)â₂ rotateswith frequency ω₁, the required pump frequency is given by ω_(p)=ω₁which is the same pump frequency reserved to engineer a linear drive onthe â₁ mode. Moreover, the term â₁â₂ ^(†)â₂ rotates at the samefrequency as those of undesired cubic terms. Hence, even if the lineardrive is realized by directly driving the phononic mode â₁, one cannotselectively drive the desired optomechanical coupling by using the pumpfrequency ω_(p)=ω₁ due to the frequency collision with the other cubicterms.

In some embodiments, to overcome these frequency collision issues, theoptomechanical coupling is realized by off-resonantly driving the term(â₁+λ)â₂{circumflex over (b)}^(†). For example, the fact that atime-dependent Hamiltonian Ĥ=λÂ{circumflex over (b)}^(†)e^(iΔt) yieldsan effective Hamiltonian Ĥ_(eff)=(x²/Δ)Â^(†)Â upon time-averaging isused assuming that the population of the {circumflex over (b)} mode issmall (e.g. {circumflex over (b)}^(†){circumflex over (b)}<<1) and thedetuning Δ is sufficiently large. Hence given a HamiltonianĤ=x(â₁+λ)â₂{circumflex over (b)}^(†)e^(−Δt)=h.c., this gives we get

${\hat{H}}_{eff} = {\frac{x^{2}\lambda}{\Delta}( {{\overset{\hat{}}{a}}_{1} + {\overset{\hat{}}{a}}_{1}^{\dagger} + \lambda + {\frac{1}{\lambda}{\overset{\hat{}}{a}}_{1}^{\dagger}{\overset{\hat{}}{a}}_{1}}} ){\overset{\hat{}}{a}}_{2}^{\dagger}{\overset{\hat{}}{a}}_{2}}$

In particular, by choosing λ=−2α, the optomechanical coupling can berealized as well as the selective frequency shift of the â₂ mode, e.g.Ĥ_(eff) ∝(â₁+â₁ ^(†)−2α)â₂ ^(†)â₂ up to an undesired cross-Ker term −â₁^(†)â₁â₂ ^(†)â₂/(2α). In this scheme, the desired selectivity isachieved because the term (â₁+λ)â₂{circumflex over (b)}^(†) is detunedfrom other undesired terms such as (â₁λ)â_(k){circumflex over (b)}^(†)with k≥3 by a frequency difference ω₂−ω_(k). Thus, the unwantedoptomechanical coupling (â₁+â₁ ^(†))â_(k) ^(†)â₂ can be suppressed by asuitable choice of the detuning Δ. It is remarked that the unwantedcross-Kerr term â₁ ^(†)â_(i)â₂ ^(†)â₂ can in principle be compensated byoff-resonantly driving another cubic term â₁â₂ {circumflex over (b)}^(†)with a different detuning Δ′≠Δ.

Lastly, similar approaches as used in the compensating Hamiltonian forthe CNOT gate can also be used for a compensating Hamiltonian for aToffoli gate.

Example Processes for Implementing an ATS-Phononic Hybrid System

FIG. 5 illustrates a process of stabilizing a nano-mechanical resonatorusing an asymmetrically-threaded superconducting quantum interferencedevice (ATS), according to some embodiments.

At block 502, a control circuit of a system comprising one or morenano-mechanical resonators causes phonon pairs to be supplied to thenano-mechanical resonator via an ATS to drive a stabilization of astorage mode of the nano-mechanical resonator such that the storage modeis maintained in a coherent state. Also, at block 504, the controlcircuit dissipates phonon/photon pairs from the nano-mechanicalresonator via an open transmission line of the control circuit that iscoupled with the nano-mechanical resonator and the ATS.

FIG. 6 illustrates a process of stabilizing multiple nano-mechanicalresonators using a multiplexed ATS, according to some embodiments.

At block 602, storage modes for a plurality of nano-mechanicalresonators that are driven by a multiplexed ATS are chosen such that thestorage modes are de-tuned. For example, block 602 may include detuningstorage modes supported by a plurality of nano-mechanical resonatorsfrom a dump resonator containing an asymmetrically-threadedsuperconducting quantum interference device At block 604 phonon pairsare supplied to a first one of the nano-mechanical resonators at a firstfrequency and at 606 phonon pairs are supplied to other ones of thenano-mechanical resonators at other frequencies such that thefrequencies for the respective storage modes of the nano-mechanicalresonators are de-tuned. For example, blocks 604 and 606 may includeapplying a pump and drive to an ATS to activate two-phonondriven-dissipative stabilization to a first one of the nano-mechanicalresonators and suppressing, via a microwave bandpass filter, correlateddecay processes from the plurality of nano-mechanical resonators.

Additionally, the storage mode frequencies and a bandwidth for a filterof the control circuit may be selected such that de-tunings of unwantedterms are larger than half the filter bandwidth. Then, at block 608 amicrowave filter with the determined filter bandwidth properties may beused to filter correlated decay terms from the plurality ofnano-mechanical resonators.

STOP Algorithm and Preparation of a Fault-Tolerant Universal Gate SetIncluding Bottom-Up Preparation of Toffoli Gates

In some embodiments, the systems described above that implement hybridacoustic-electrical qubits may be used to implement a universal gateset. In some embodiments, error correction may be used to correct forerrors and/or noise in such systems. In some embodiments, a STOPalgorithm, as described herein, may provide an efficient protocol forproviding error detection and/or correction. In some embodiments,systems, as described above, that implement hybrid acoustic-electricalqubits may introduce noise that is biased towards phase flip errors.With such knowledge about error bias, error correction protocols, suchas a STOP algorithm, may be used to efficiently correct for errors.Additionally, as further discussed below, error correction may be usedto correct for errors when preparing Toffoli gates using a bottom-upapproach (and/or when using a top-down approach which is furtherdiscussed in the next section).

In some embodiments, a STOP algorithm may be used to determine when itis acceptable to STOP measuring stabilizer measurements as part of anerror detection/error correction operation while guaranteeing a lowprobability of logical errors. For example, a STOP algorithm may be usedto measure stabilizer measurements prior to performing a Toffoli gatewherein measured errors are corrected prior to applying the Toffoligate.

An alternative to using a STOP decoder may be to use graph based errorcorrection techniques. However, these techniques are typicallypredicated on the use of Clifford gates and are not as useful whenapplying Toffoli gates. For example, these techniques involve measuringdata qubits at the end of performing an operation to determine errors.However, for non-Clifford gates, a single qubit error of the initialinput qubits can cause a logical failure that may not be detected usinga standard graph based error correction technique.

In contrast, a STOP algorithm measures stabilizers for input data qubitssuch that error detection and/or correction can be performed prior toperforming an operation, such as a non-Clifford gate. In addition,instead of measuring the stabilizers for the data qubits a fixed numberof times, which may be insufficient to detect/correct logical errors insome situations, or which may be unnecessary in other situations, a STOPalgorithm may be used to determine when stabilizer measurements can bestopped while still guaranteeing a low probability of logical errors.For example, in some embodiments, a STOP algorithm may guarantee that atotal number of failures is less than a code distance of repeatedlyencoded data qubits (e.g. a repetition code) divided by two. Thus themajority of the repeated data qubits are known to not be erroneous and alogical error will not occur because the majority of the encoded dataqubits are correct. For example, errors can be tolerated as long as thetotal number of errors is less than the code distance divided by two. Insuch situations, the errors will not result in a logical error, becausethe majority of the encoded qubits are not erroneous. Note that aphysical error is distinct from a logical error. A physical error actson an individual qubit, whereas a logical error is an erroneous logicaloutput determined based on physical qubits. A logical error cannot bedirectly detected, and if not detected, cannot be corrected. Forexample, an uncorrected physical error may result in a logical error,but if the physical error was undetected, there is no way tosubsequently measure the logical error caused by the physical error,without knowing about the physical error.

In some embodiments, a STOP algorithm may also be applied to qubits usedfor performing non-Clifford gates, such as a Toffoli gate. Also, in someembodiments, a STOP algorithm may be used when growing a repetition codefrom a first code distance to a second code distance, whereinstabilizers at a boundary between code blocks that are being joined togrow the repetition code are measured. The STOP algorithm may be used todetermine when repeated measurements of the stabilizers at the boundarycan be stopped without introducing logical errors into the expandedrepetition code.

In some embodiments, when preparing a Toffoli gate, a STOP algorithm maybe used to detect and/or correct errors in the initial computationalbasis states used to prepare the Toffoli gate. The STOP algorithm mayalso be used in preparing Clifford gates that are applied in a sequenceto implement the Toffoli gate, wherein the STOP algorithm is used todetect/correct errors in the Clifford gates. Additionally, the STOPalgorithm may be used to perform error detection/correction betweenmeasurements of g_(A) which is repeatedly measured as part of preparingthe Toffoli gates using a bottom up approach, as further discussedbelow. In some embodiments, a round of error detection may be performedbetween each round of measuring g_(A).

In some embodiments, a STOP algorithm may follow an algorithm similar tothe algorithm shown below:

-   -   Set: t=(d−1)/2; n_(diff)=0; countSyn=1; SynRep=1

-   n_(diff) Increase=0; test=0

-   while test=0 {

-   if n_(diff)=t{

-   test=1;

-   }    Measure the error syndrome. Store the error syndrome from the    previous round in synPreviousRound and the current syndrome in    synCurrentRound.

-   if (countSyn>1) {

-   if (synPreviousRound=synCurrentRound) {

-   SynRep=SynRep+1;

-   n_(diff) Increase=0;

-   } else {

-   SynRep=0;

-   if (n_(diff) Increase=0) {

-   n_(diff)=n!″##+1;

-   n_(diff) Increase=1;

-   } else

-   n_(diff) Increase=0

-   }

-   }

-   if (SynRep=t−n_(diff)+1) {

-   test=1;

-   }

-   countSyn=countSyn+1;

-   }

Said another way, let S_(j) be the error syndrome of the j^(th) round ofsyndrome measurements. The goal of the STOP algorithm is to compute theminimum number of faults that can cause changes between two consecutivesyndromes. The worst case scenario is where a single two-qubit gatefailure results in three different syndrome outcomes. To see this, letS_(k−1) be the syndrome from round k−1. Now suppose the operator X^(⊗n)is measured using the circuit 700 shown in FIG. 7 with the input errorE_(in) such that s(E_(in))=s_(k−1). Further, suppose the last two-qubitgate fails resulting in the error X⊗. The X error results in the dataerror XE_(in) (e.g. data error 702) which may have syndrome s_(k−1),while the Z error flips the syndrome outcome (e.g. measurement outcome704), resulting in the syndrome s_(k) which can be different froms_(k−1) and s_(k+1). Hence without any other failure, this example showsthat a single fault can cause three distinct syndromes s_(k−1), s_(k)and s_(k+1)

The STOP decoder tracks consecutive syndrome measurement outcomes s₂,s₂, . . . , s_(r)′, where r is the total number of syndrome measurements(r is not fixed), between two syndrome measurement rounds k and k+1(with corresponding syndromes s_(k) and s_(k+1)), wherein the minimumnumber of faults causing a change in syndrome outcome (represented bythe variable n_(diff)) is only incremented if n_(diff) did not increasein round k.

Now assuming there were no more than t=(d−1)/2 faults for a distance derror correcting code, if the same syndrome s_(j) was repeatedt−n_(diff)+1 times in a row, then the syndrome must have been correct(i.e. there were no measurement errors). As such, in this situation onecould use the syndrome s_(j) to correct the errors and terminate theprotocol.

Similarly, if n_(diff)=t, then there must have been at least t faults.As such, by repeating the syndrome measurement one more time (resultingin the syndrome s_(r)) and using that syndrome to decode, there wouldneed to be more than t faults for s_(r) to produce the wrong correction.Hence the STOP decoder terminates if one of the following two conditionsare satisfied:

-   -   1) The syndrome s_(j) is obtained t−n_(diff)+1 times in a row.        In which case the s_(j) syndrome is used to decode. OR    -   2) The variable n_(diff) gets incremented to n_(diff)=t. In        which case, the syndrome measurement is repeated one more time        and the repeated syndrome measurement is used to decode.        Stabilizer Operations with the Repetition Code

In some embodiments, logical computational basis states may be preparedusing a repetition code. In some embodiments, stabilizer measurements ofa repetition code may be performed using a STOP algorithm, as describedabove. Also, in some embodiments, the methods described herein may beapplied to any family of Calderbank-Shor-Steane (CSS) codes.

In some embodiments, using the fact that for an n-qubit repetitioncode|+>_(L)=|+

^(⊗n), preparing |+

^(⊗n) followed by a logical Z_(L)=Z^(⊗n) measurement (see FIG. 8)projects the state to |0

_(L) given a+1 outcome and |1

_(L) given a −1 outcome. Since a measurement error on the ancillaresults in a logical X_(L)=X₁ error applied to the data, fault tolerancecan be achieved by repeating the measurement of Z₁ using the STOPalgorithm (where the syndrome corresponds to the ancilla measurementoutcome), and applying the appropriate X_(L) correction given the finalmeasurement outcome. For instance, if |0

_(L) is the desired state and the final measurement outcome at thetermination of the STOP algorithm is −1, X₁ would be applied to thedata. Lastly, note that only X errors can propagate from the ancilla tothe data but are exponentially suppressed by the cat-qubits.

In some embodiments, computational basis states may be prepared using anapproach that only involves stabilizer measurements. For example,starting with the state |ψ

₁=|0

^(⊗n) which is a+1 eigenstate of Z_(L), measure all stabilizers of therepetition code (each having a random±1 outcome) resulting in the state:

$ \psi \rangle_{2} = {\prod\limits_{i = 1}^{n - 1}{( \frac{I \pm {X_{i}X_{i + 1}}}{2} ) 0 \rangle^{\otimes n}}}$

If the measurement outcome of X_(k)X_(k+1) is −1, the correction Π_(j=1)^(k)Z₁ can be applied to the data to flip the sign back to +1. Howevergiven the possibility of measurement errors, the measurement of allstabilizers (X₁X₂, X₂X₃, . . . , X_(n−1)X_(n)) must be repeated. Ifphysical non-Clifford gates are applied prior to measuring the data,then the STOP algorithm can be used to determine when to stop measuringthe syndrome outcomes. Subsequently, minimum-weight perfect matching(MWPM) may be applied to the full syndrome history to correct errors andapply the appropriate Z corrections to fix the code-space given theinitial stabilizer measurements. When Clifford gates are applied to thedata qubits in order to prepare a |TOF

magic state, this second scheme for preparing the computational basisstates may be used along with the STOP algorithm.

Additionally, it is pointed out that although the logical component ofan uncorrectable error E^((z))Z_(L) (where E^((z)) is correctable) canalways be absorbed by |0

_(L) resulting in an output state |ψ

_(out)=E^((z)|)0

_(L), it is still important to have a fault-tolerant preparation schemefor |0

_(L) (and thus to repeat the measurement of all stabilizers enoughtimes). For instance, if a single fault results in a weight-twocorrectable Z error (assuming n≥5), a second failure during a subsequentpart of the computation can combine with the weight-two error resultingin an uncorrectable data qubit error. Hence, such a preparation protocolwould not be fault-tolerant up to the full code distance.

Implementation of Logical Clifford Gates

Since the CNOT gate is transversal for the repetition code, focus can beplaced on implementing a set of single qubit Clifford operations. Recallthat the Clifford group is generated by:

${P_{n}^{(2)} = \langle {H_{i},S_{i},{CNOT}_{ij}} \rangle},{{{where}\mspace{14mu} H} = {\frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1 \\1 & {- 1}\end{pmatrix}}},{S = \begin{pmatrix}1 & 0 \\0 & i\end{pmatrix}}$

Note that H and S given above are the Hadamard and phase gate operators.In some embodiments, S and Q=SHS may form a generating set forsingle-qubit Clifford operations. In implementing such states, injectionof the state

${{ i \rangle} = {\frac{1}{\sqrt{2}}( { 0 \rangle + {i 1 \rangle}} )}},$which is a +1 eigenstate of the Pauli operator, may be performed.

In FIG. 10 a circuit 1000 for implementing S_(L) is given, wherein thecircuit takes |i

_(L) as an input state and includes a transversal CNOT gate and alogical Z-basis measurement. If a −1 measurement outcome is obtained, aZ_(L) correction is applied to the data. Note however that a measurementerror can result in a logical Z_(L) being applied incorrectly to thedata. As such, to guarantee fault-tolerance, one can repeat the circuitof FIG. 10 and use the STOP algorithm to decide when to terminate. Thefinal measurement outcome may then be used to determine if Z_(L)correction is necessary. The implementation of S can thus be summarizedas follows:

-   -   1.) Implement the circuit shown in FIG. 10 and let the        measurement outcome be S₁;    -   2.) Repeat the circuit of FIG. 10 and use the STOP algorithm to        decide when to terminate; and    -   3.) If the final measurement outcome S_(r)=+1, do nothing,        otherwise apply Z_(L)=Z₁Z₂ . . . Z_(n) to the data.

The circuit 900 for implementing the logical Q=SHS gate is given in FIG.9. The circuit consists of an injected |i

_(L) state, a transversal CNOT gate, and a logical X-basis measurementis applied to the input data qubits. If the measurement outcome is −1,Y_(L) is applied to the data. As with the S gate, the application of thecircuit in FIG. 9 is repeated according to the STOP algorithm to protectagainst measurement errors. The full implementation of Q_(L) is given asfollows:

-   -   1.) Implement the circuit in FIG. 9 and let the measurement        outcome be S₁;    -   2.) Repeat the circuit in FIG. 9 and use the STOP algorithm to        decide when to terminate; and    -   3.) If the final measurement outcome S_(r)=+1, do nothing,        otherwise apply Y_(L)=Y₁Z₂ . . . Z_(n) to the data.

Note that the logical Hadamard gate can be obtained from the S_(L) andQ_(L) protocols using the identity H=S^(†)SHSS^(†)=S^(†)QS^(†). Henceignoring repetitions of the circuits in FIGS. 9 and 10, theimplementation of H_(L) requires three logical CNOT gates, two |−i

_(L) and one |i

_(L) state, two logical Z basis measurements and one logical X basismeasurement. Instead of using two logical Hadamard gates and one CNOTgate to obtain a CZ gate, a more efficient circuit is shown in FIG. 12.Lastly, since the circuits in FIGS. 9 and 10 contain only stabilizeroperations and injected |i

_(L) states, using the STOP algorithm to repeat the measurements is notstrictly necessary. For instance, one could repeat the measurement afixed number of times and majority vote instead of using the STOPalgorithm. However in low noise rate regimes, the STOP algorithm canpotentially be much more efficient since the average number ofrepetitions for the measurements can approach t+1 where t=(d−1)/2.

Growing Encoded Data Qubits to Larger Code Distances with the RepetitionCode

In some embodiments, a state |ψ

_(d1)=α|0

_(d1)+β|1

_(d1) encoded in a distance d₁ repetition code is grown to a state |ψ

_(d2)=α|0

_(d2)+β|1

_(d2) encoded in a distance d₂ repetition code. Such a protocol may beused to grow |TOF

magic states as further described below.

Let S_(d1)=

X₁X₂, X₂X₃, . . . X_(d) ₁ ⁻¹X_(d)

be the stabilizer group for a distance d₁ repetition code withcardinality |S_(d1)|=d₁−1. Similarly S_(d′) ₁ is defined as S_(d′) ₁ =

X_(d) ₁ ₊₁X_(d) ₁ ₊₂, . . . X_(d) ₂ ⁻¹X_(d2)

with |S_(d′) ₁ |=d₂−d₁−1. Furthermore the stabilizer group for adistance d₂ repetition code is given by S_(d2)=

X₁X₂, X₂X₃, . . . X_(d) ₂ ⁻¹X_(d) ₂

.

Also g_(i) ^((d1)) is defined as the i'th stabilizer in S_(d1) and g_(i)^((d′1)) to be the i′'th stabilizer in S_(d′) ₁ , so that g_(i) ^((d) ¹⁾=X_(i)X_(i+1) and g_(i) ^((d′) ¹ ⁾=X_(d) ₁ _(+i)X_(d) ₁ _(+i+1). Theprotocol for growing |ψ

_(d1) to |ψ

_(d2) is given as follows:

-   -   1.) Prepare the state |ψ₁        =|0        ^(⊗(d) ² ^(−d) ¹ ⁾.    -   2.) Measure all stabilizers in S_(d′) ₁ resulting in the state

$ \psi_{2} \rangle = {\prod\limits_{i = 1}^{d_{2} - d_{1} - 1}{( \frac{I \pm g_{i}^{(d_{1}^{\prime})}}{2} ){ 0 \rangle^{\otimes {({d_{2} - d_{1}})}}.}}}$

-   -   3.) Repeat the measurement of stabilizers in S_(d′) ₁ using the        STOP algorithm and apply MWPM to the syndrome history to correct        errors and project to the code-space. If g_(i) ^((d′) ¹ ⁾ is        measured as −1 in the first round, apply the correction Π_(k=d)        ₁ ₊₁ ^(d) ¹ Z_(k) to the data.    -   4.) Prepare the state |ψ₃        =|ψ_(d) ₁        ⊗|ψ₂        and measure X_(d) ₁ X_(d) ₁ ₊₁.    -   5. Repeat the measurement of all stabilizers of S_(d2) using the        STOP algorithm and use MWPM over the syndrome history to correct        errors. If in the first round the stabilizer X_(d) ₁ X_(d) ₁ ₊₁        is measured as −1, apply the correction Π_(i=1) ^(d) ¹ Z_(i).

The growing scheme involves two blocks, the first being the state|ψ>_(d1) which is grown to |ψ

_(d2). The second block involves the set of qubits which are prepared inthe state |ψ

_(d′) ₁ and stabilized by S_(d′) ₁ (steps 1-3). The key is to measurethe boundary operator X_(d) ₁ X_(d) ₁₊₁ between the two blocks whicheffectively merges both blocks into the encoded state |ψ

₂, which is a simple implementation of lattice surgery. To see this,consider the state prior to step 4:

$\begin{matrix}{ \psi \rangle_{3} = { \overset{¯}{\psi} \rangle_{d1} \otimes  \psi_{2} \rangle_{d_{1}^{\prime}}}} \\{= {{\alpha{ 0 \rangle_{d1} \otimes  \psi_{2} \rangle_{d_{1}^{\prime}}}} + {\beta{ 1 \rangle_{d1} \otimes  \psi_{2} \rangle_{d_{1}^{\prime}}}}}} \\{= {{\alpha{\prod\limits_{i = {d_{1} + 1}}^{d_{2} - 1}{( \frac{I + g_{i}^{(d_{1}^{\prime})}}{2} ){ 0 \rangle_{d1} \otimes  0 \rangle^{\otimes {({d_{2} - d_{1}})}}}}}} +}} \\{{\beta X}_{1}{\prod\limits_{i = {d_{1} + 1}}^{d_{2} - 1}{( \frac{I + g_{i}^{(d_{1}^{\prime})}}{2} ){ 0 \rangle_{d1} \otimes  0 \rangle^{\otimes {({d_{2} - d_{1}})}}}}}}\end{matrix}$

In the above equation |1

_(d1)=X₁|0

_(d1). Also, when measuring X_(d) ₁ X_(d) ₁₊₁ and performing thecorrection Π_(i=1) ^(d) ¹ Z₁ if the measurement outcome is −1, |ψ

₃ is projected to:

$\begin{matrix}{ \psi_{f} \rangle = {{\prod\limits_{i = {d_{1} + 1}}^{d_{2} - 1}{( \frac{I + g_{i}^{(d_{1}^{\prime})}}{2} )( \frac{I + {X_{d_{1}}X_{d_{1 + 1}}}}{2} ){\prod\limits_{j = 1}^{d_{1} - 1}{( \frac{I + g_{i}^{(d_{1})}}{2} ) 0 \rangle^{\otimes d_{2}}}}}} +}} \\{\beta\; X_{1}{\prod\limits_{i = {d_{1} + 1}}^{d_{2} - 1}{( \frac{I + g_{i}^{(d_{1}^{\prime})}}{2} )( \frac{I + {X_{d_{1}}X_{d_{1 + 1}}}}{2} ){\prod\limits_{i = {d_{1} + 1}}^{d_{2} - 1}{( \frac{I + g_{ij}^{(d_{1})}}{2} ) 0 \rangle^{\otimes d_{2}}}}}}} \\{= {{\alpha\;{\prod\limits_{i = 1}^{d_{2} - 1}{( \frac{I + {X_{i}X_{i + 1}}}{2} ) 0 \rangle^{\otimes d_{2}}}}} + {\beta\; X_{1}{\prod\limits_{i = 1}^{d_{2} - 1}{( \frac{I + {X_{i}X_{i + 1}}}{2} ) 0 \rangle^{\otimes d_{2}}}}}}} \\{= {{\alpha 0 \rangle_{d_{2}}} + {\beta\; X_{1} 0 \rangle_{d_{2}}}}} \\{=  \psi \rangle_{d_{2}}}\end{matrix}$

The rounds of repeated stabilizer measurements in steps 3 and 5 (above)may be required due to the random outcomes and measurement errors whichcan occur when performing the appropriate projections. A pictorialrepresentation for the growing scheme is shown in FIG. 16.

Bottom-Up Fault Tolerant Preparation of the |TOF

Magic State

In some embodiments, a |TOF

magic state can be prepared using the repetition code, wherein the |TOF

magic state is used in simulating a Toffoli gate.

The |TOF

magic state is given by:

${ {TOF} \rangle = {\frac{1}{2}{\sum\limits_{x_{1},x_{2}}{ X_{1} \rangle X_{2} \rangle {X_{1} ⩓ X_{2}} \rangle}}}},$which is stabilized by the Abelian groupS _(TOF) =

g _(A) ,g _(B) ,g _(C)

whereg _(A) =X ₁ CNOT_(2,3)g _(B) =X ₂ CNOT_(1,3)g _(C) =Z ₃ CZ _(1,2)

Given one copy of a |TOF

magic state, a Toffoli gate can be simulated using the circuit 1102 inFIG. 11A, and the required Clifford corrections are given in FIG. 11B.Note that if a correction involves the stabilizer g_(C), the CZ gate canbe implemented using the circuit 1200 in FIG. 12. Also, note that forthe Clifford corrections a 0 indicates a +1 measurement outcome whereasa 1 indicates a −1 measurement outcome (in either the X or Z basis). Thestabilizers g_(A), g_(B) and g_(C) are given in the equations above.

Next, how to fault-tolerantly prepare the |TOF

magic state is discussed. First, note that the state

$ \psi_{1} \rangle = {\frac{1}{\sqrt{2}}( { 010 \rangle +  111 \rangle} )}$is stabilized by g_(A) and g_(C). Such a state can straightforwardly beprepared using the circuit 1300 in FIG. 13. In what follows, physicalToffoli gates will need to be applied between ancilla qubits and |ψ₁

prior to measuring the data. As such, it is important that the states |0

_(L) and |1

_(L) in the circuit 1300 of FIG. 13 be prepared using the STOP algorithmsince otherwise measurement errors in the last ancilla measurement roundcould lead to logical failures. Once |+

_(L), |1

_(L) and |0

_(L) have been prepared, the CNOT gate 1302 in FIG. 13 is appliedtransversally.

Now, given a copy of |ψ₁

, the |TOF

magic state can be prepared by measuring g_(A) using the circuit 1400 ofFIG. 14 resulting in the state |ψ

_(out). If the measurement outcome is +1, |ψ

_(out)=|TOF

. Otherwise, if the measurement is −1, |ψ

_(out)=Z₂|TOF

. Hence given a −1 measurement outcome, a logical Z_(L) correction isapplied to the second code block. A more detailed implementation 1700 ofthe controlled-g_(A) gate 1400 is shown in FIG. 17. For example, thecircuit 1700 is shown for measuring a code with code distance d=3. Ingeneral, d Toffoli gates are required. Note that for the repetitioncode, a single CNOT gate is required since X_(L)=X₁. Further, due to thetransversal CNOT gates, physical Toffoli gates are applied sequentiallyas shown in the figure. Note that such a circuit can be used for anyCalderbank-Shor-Steane (CSS) code. The sequence of Toffoli gates wouldremain unchanged. Generally more two-qubit gates would be requireddepending on the minimal weight representation of X_(L).

Note that since the CNOT_(1,3) gate can be done transversally for therepetition code, and that X_(L) on the second code block is given by aphysical X gate on the first qubit of that block, the controlled-g_(A)circuit can be highly parallelized thus greatly simplifying itsimplementation. For example, FIG. 18 illustrates a more parallelizedcircuit for measuring g_(A) which requires one flag qubit 1802. Such acircuit reduces the depth of Toffoli gates in half at the cost of addingto time-steps due to the extra CNOT gates. The flag qubit can also beused for detecting X errors arising on the control qubits of the CNOTand Toffoli gates. If an X error occurs, the flag qubit measurementoutcome will be −1 instead of +1. As in FIGS. 14, 15, and 17, if eitherthe X or Z basis measurement outcomes are −1 instead of +1, the entire|TOF

magic state preparation protocol is aborted and begins anew.

As was also the case as discussed above with regard to the repetitioncode, a measurement error on the ancilla results in a logical Z₂ failureand thus the measurement of g_(A) needs to be repeated. This can be donedeterministically using the STOP algorithm. However due to theincreasing circuit depth with increasing repetition code distance inaddition to the high cost of the controlled-g_(A) gate, such a schemedoes not have a threshold and results in relatively high logical failurerates. An alternative approach is to use an error detection scheme byrepeating the measurement of g_(A) exactly (d−1)/2 times for a distanced repetition code. In between each measurement of g_(A), one round oferror detection is applied to the data qubits by measuring thestabilizers of the repetition code. This is shown in FIG. 15. If any ofthe measurement outcomes are non-trivial, the protocol for preparing the|TOF

magic state is aborted and reinitialized. In FIG. 19A, an example 1900is provided of a two-dimensional layout of qubits and sequence ofoperations for measuring g_(A), which is compatible with the abovedescribed ATS architecture for a distance 5 repetition code. Such alayout uses a minimum number of ancilla qubits and can bestraightforwardly generalized to arbitrary repetition code distances.The ancilla qubits are used to first prepare a GHZ state. Subsequentlythe required Toffoli and CNOT gates are applied, followed by adisentangling of the GHZ states and measurement of the |+

state ancilla. The equivalent circuit 1950 implementing the g_(A)measurement for a d=5 repetition code is shown in FIG. 19B.

Notice that to respect the connectivity constraints of the ATS's, thelighter grey vertices 1902 need to be swapped with the darker greyvertices 1904 on the second block (shown in the upper left corner of thelattice of FIG. 19A). Such a role reversal between the ancilla and dataqubits does not lead to additional cross-talk errors for the reasonsdiscussed above with regard to a multiplexed ATS with filtering and thuscan be tolerated. As such, all controlled g_(A) measurements in FIG. 15may be implemented using the circuit 1950 in FIG. 19B with the qubitlayout given in FIG. 19A.

Lastly, note that circuit 1950 in FIG. 19B used to measure g_(A) is notfault-tolerant to X or Y errors. However, since it is assumed that X andY errors are exponentially suppressed, flag qubits for detecting X-typeerror propagation are unnecessary as long as X or Y error ratesmultiplied by the total number of fault locations are below the targetlevels for algorithms of interest.

FIG. 20A illustrates high-level steps of a protocol for implementing aSTOP algorithm, according to some embodiments.

At block 2002 syndrome outcome measurements are performed for anarbitrary Calderbank-Shor-Steane code. At block 2004 consecutive ones ofthe syndrome outcomes are tracked to generate a syndrome history. Atblock 2006 syndrome measurements are stopped if condition 1 (shown inblock 2006A) or condition 2 (shown in block 2006B) are met. Conditionone is that a same syndrome outcome is repeated a threshold number oftimes in a row, wherein the threshold is equal to ((d−1)/2)−n_(diff).−1Condition two is that n_(diff) is equal to (d−1)/2, and one additionalsyndrome outcome has been measured subsequent to reachingn_(diff)(d−1)/2. If either of these conditions are met, then themeasurements of the syndrome outcomes can be stopped. At block 2008 ifcondition one is met, the repeated syndrome outcome is used to performerror correction. Also, at block 2008 if condition two is met, thesubsequently measured syndrome outcome is used to perform errorcorrection.

FIG. 20B illustrates high-level steps for determining a parameter (miff)used in the STOP algorithm, according to some embodiments.

At block 2052 n_(diff) is initialized with an initial value equal tozero. At block 2054 a first round of syndrome outcome measurements isperformed. Also, at block 2056, a second round of syndrome outcomemeasurements is performed. At block 2058 it is determined if thesyndrome outcomes measured in the round performed at block 2056 (e.g.the current round of syndrome outcomes) differ from the syndromeoutcomes measured for the preceding round. If so, at block 2060 it isdetermined if n_(diff) was incremented in the previous round, if notn_(diff) is incremented by one at block 2062 and the process repeats fora subsequent round of syndrome outcome measurements. However, note thatwhen condition one or condition two (as shown in blocks 2006A and 2006B)are met, the syndrome measurements are stopped. If the syndrome outcomesmeasured in the round performed at block 2056 (e.g. the current round ofsyndrome outcomes) are the same as the syndrome outcomes measured forthe preceding round or it is determined at block 2060 if n_(diff) wasincremented for the preceding round, the process reverts to block 2056and another round of syndrome outcomes are measured without incrementingn_(diff).

FIG. 21 illustrates high-level steps of a protocol for growing arepetition code from a first code distance to a second code distanceusing a STOP algorithm, according to some embodiments.

At block 2102 a |ψ₁

state is prepared as described above, for example using the circuitshown in FIG. 13. At block 2104, all stabilizers S_(d′1) are measuredresulting in a state |ψ₂

. This may be done as described above with regard to stabilizeroperations for the repetition code. At block 2106, the measurements ofthe stabilizers in S_(d′1) are repeated using the STOP algorithm andMWPM is applied to the syndrome history to correct errors and projectthe code into the increased code space. At block 2108 a |ψ₃

state is prepared and X_(d1)X_(d1+1) are measured. At block 2110 themeasurements of all the stabilizers of S_(d2) are repeated using theSTOP algorithm and MWPM is applied over the syndrome history to correcterrors.

FIG. 22 illustrates high-level steps of a protocol for implementing alogical Toffoli gate using a bottom-up approach with Toffoli magic stateinjection, according to some embodiments.

At block 22, fault-tolerant computational basis states are preparedusing the STOP algorithm, wherein the fault-tolerant computational basisstates are to be used as inputs for a Toffoli gate preparation. Atblock, 2204, a CNOT gate is transversally applied to the fault-tolerantcomputational basis states to prepare a |ψ₁

state. At block 2206 g_(A) is measured for the |ψ₁

state, which yields a state |ψ_(out)

. If the measurement of g_(A) has a measurement outcome of −1 then a Zcorrection is applied. This projects the |ψ₁

state into a |TOFF

state. At block 2208 the measurements of g_(A) are repeated such thatg_(A) is measured (d−1)/2 times. Between rounds of measurement of g_(A),error detection is performed. If non-trivial values are measured foreither g_(A) or the error detection, the protocol is aborted andre-initiated anew. At block 2208, if all the measurements outcomes ofg_(A) and the error detection performed at 2208 are trivial, a Toffolimagic state (e.g. |TOFF

state) is prepared based on the measurement of g_(A) and the state|ψ_(out)

. For example, if all the measurements outcomes of g_(A) and the errordetection performed at 2208 are trivial, then |ψ_(out)

=|TOFF

. At block 2210 a sequence of Clifford gates as shown in circuit 1102 ofFIG. 11A are applied. Also the Clifford error corrections shown in FIG.11B are applied. This may be done as part of a top down distillation ofa logical Toffoli gate (as described in more detail below) that utilizesthe prepared Toffoli magic state as an input to the distillationprocess.

Top-Down Distillation Process to Yield Low-Error Rate Toffoli Gates

As discussed above, the Toffoli gate when combined with the Cliffordgroup forms a universal gate set for quantum computation. Alternatively,universality can be achieved by complementing the Clifford group with asupply of high-fidelity Toffoli magic states encoded in a suitablequantum error correction code. For many high threshold error correctioncodes, such as repetition (for very biased noise) or surface codes, highfidelity Toffoli magic states are difficult to prepare. The paradigm ofmagic state distillation uses encoded Clifford operations to distillhigher fidelity magic states from lower fidelity magic states. Forexample, the Toffoli magic states prepared using the bottom-up approachdescribed above may be used as in a magic state distillation process toyield even lower fault-rate Toffoli magic states.

The conventional approach to magic state distillation uses a supply oflow fidelity T magic states as inputs to protocols that output othertypes of magic state, including TOFF states. However, in somearchitectures the supply of noisy TOFF states can be prepared at betterfidelity than the noisy T states. This is because allCalderbank-Shor-Steane (CSS) codes, such as surface and repetitioncodes, have a transversal CNOT and this property can be used to robustlyprepare the TOFF state (as described above for bottom-up approach).However, the success probability of such “bottom-up preparation”protocols drops as the target fidelity is increased and so it isdesirable to design magic distillation protocols that can further purifynoisy TOFF states at low overhead. If the bottom-up TOFF protocol isused to prepare TOFF states with 10⁻⁵-10⁻⁶ error rates, then for severalquantum algorithms only a single round of magic state distillation wouldbe need to achieve 10⁻⁹-10⁻¹⁰ logical error rates. In contrast, for Tstates prepared at 10⁻³-10⁻⁴ error rates, to achieve comparable logicalerror rates would require two rounds of magic state distillation withquadratic error suppression, or alternatively a single round of the 15T→1 T protocol with low ( 1/15) rate.

In some embodiments, to address these uses a top-down distillationprocess is performed that uses TOFF or CCZ states without using any Tstates, either as raw distillation material or as catalysts. Alsotriorthgonal codes are not used in the usual sense, but instead providea new technique for protocol design by describing CCZ circuits in termsof cubic polynomials. It is noted that CCZ states are Cliffordequivalent to TOFF states, and when using cubic polynomial formalism, itwill be beneficial to work in the language of CCZ states. As an exampleof these techniques, it is shown that it is possible to achieve8CCZ→2CCZ distillation, equivalently 8TOFF→2TOFF detecting a fault onany single TOFF state. In cases where noise on the CCZ state is verybiased towards certain types of faults, more compact and efficientprotocols are possible, which are also described.

In some embodiments, various architectures may be used to implement thedistillation processes described herein, such as a 2D architecture usingthe repetition code, asymmetric surface codes (for biased noise) orconventional square surface codes. The 2D implementation performs therequired Clifford operations using lattice surgery to realize a suitablesequence of multi-qubit Pauli observables (also called multi-patchmeasurements).

FIG. 23 illustrates high-level steps for distilling a low-error ratelogical Toffoli gate using multiple ones of the logical Toffoli gatesprepared using a bottom-up approach as described in FIG. 22, accordingto some embodiments.

For example, at 2302 physical Toffoli magic states are generated, whichmay have a probability of error of approximately 2.8×10⁻⁴. This errorprobability may be improved by an order of magnitude or more by applyingthe STOP algorithm and error correction techniques described above forthe bottom-up approach. For example, block 2304 illustrates theimprovements in error-rate that are realized by utilizing the bottom-upapproach. However, further improvements in error rate can be achieved byperforming a top-down distillation process. For example, block 2306illustrates that error probabilities may be reduced to approximately8×10⁻¹⁰ by performing a single round of distillation using Toffoli magicstates prepared using the bottom up approach as inputs.

FIG. 24 illustrates a layout of multiple bottom up Toffoli gates thatare used to distill low-error rate logical Toffoli gates, according tosome embodiments.

To give a general view of the distillation process, FIG. 24 illustratesa circuit 2400 that includes qubits that have been prepared to implementbottom up (e.g. “BU”) magic states. Also other qubits of the circuithave been prepared to implement CCZ magic states (or low-error rateToffoli magic states/gates). Additionally, some of the qubits implementan error check for the CCZ magic states. For example each set of checkqubits may be associated with a pair of CCZ magic states.

Synthesis

First, observe that a CCZ_(i,j,k) gate on qubits i, j and k, willperform:CCZ _(i,j,k) |x ₁ ,x ₂ ,x ₃ , . . . ,x _(n))=(−1)^(x) ^(i) ^(x) ^(j)^(x) ^(k) |x ₁ ,x ₂ ,x ₃ , . . . ,x _(n)

where |x

=|x₁, x₂, x₃, . . . , x_(n)

represents a computational basis state described as a binary stringx=(x₁, x₂, x₃, . . . , x_(n)

. More generally, consider conjugating these CCZ gates with a CNOTcircuit. For any invertible matrix J, there exists a CNOT circuit V suchthat:V=Σ _(x) |x

Jx|.

Composing these operations a generalized CCZ gate is given by:CCZ

V ^(†) CCZ _(1,2,3) V|x)=(−1)^((J) ¹ ^(X)(J) ² ^(X)(J) ³ ^(X)) |x

where J_(k) is the kth column vector of J andJ_(k)x=Σ_(α)[J_(k)]_(α)x_(α) is the dot product between this vector andthe bit string vector x. Because J is invertible, the J_(k) must belinearly independent, but otherwise there are no constraints.Furthermore, only three column vectors are needed to describe the actionof a single generalized CCZ gate.

Alternatively, a generalized CCZ gate can be realized using a single CCZmagic state as shown in 2504 FIG. 25. The CCZ magic state is:

$ {CCZ} \rangle = {{{CCZ}  + \rangle  + \rangle  + \rangle} = {\frac{1}{2\sqrt{2}}{\sum\limits_{y \in {\mathbb{z}}_{2}^{3}}{( {- 1} )^{y_{1}y_{2}y_{3}} y_{1} \rangle y_{2} \rangle y_{3} \rangle}}}}$and it can be used to inject a CCZ gate as illustrated in FIG. 25 andwhich can be extended to generalized CCZ gates by controlling the CNOTgates determined by the associated vectors J₁, J₂, and J₃. Furthermore,the CNOTs in the CCZ injection can be replaced with a sequence ofmulti-qubit Pauli measurements, which are the primitive operations inlattice surgery based architectures.

In some embodiments, a unitary as shown below can be composed using CCZ,CZ, Z and CNOT gates:U=Σ _(x)(−1)^(ƒ(x)) |Jx

x|where J is invertible and ƒ:

₂ ^(n)→

₂ is some Boolean function expressible as a cubic polynomial. Formally,this can be expressed as shown below in Theorem 1:

Theorem 1: Let U be a unitary of the form of the equation above with afunction ƒ such that there exists a cubic polynomial representation:

${f(x)} = {\sum\limits_{a \leq b \leq c}{F_{a,b,c}x_{a}x_{b}x_{c}}}$with integers F_(i,j,k). It follows that there are many differentfactorizations of the polynomial as follows:

${f(x)} = {{\sum\limits_{j = 1}^{\zeta}{( {J_{1}^{j} \cdot x} )( {J_{2}^{j} \cdot x} )( {J_{3}^{j} \cdot x} )}} + {x^{T}Qx}}$where J_(k) ^(i) are binary vectors (and therefore linear functions) anda Q is a lower-triangular binary matrix (representing a quadraticBoolean function). Then there exists a circuit composed of {CCZ, CZ, Z,CNOT} that implements U using at most copies of the CCZ gates. We callthe minimal such the cubic rank of the polynomial.CCZ Magic State Distillation

In some embodiments, cubic polynomial formalism is used to developroutines for distillation of high-fidelity |CCZ

magic states. For example, given a supply noisy |CCZ

states with Z noise, the noisy |CCZ

states can be distilled using Clifford operations to obtain a smallernumber of |CCZ

states with less noise. Note that given any noise model, |CCZ

magic states can be twirled so that the noise becomes pure Z noise.Accordingly, in some embodiments, a circuit is designed to realize atarget unitary, say U=|CCZ

^(⊗k) that acts on 3 k qubits plus some number m of check qubits.However, instead of minimizing the number of CCZ gates in the circuit,the proposed design is such that Z errors on the |CCZ

magic state propagate onto the check qubits. Therefore, by measuring thecheck qubits at the end of the circuit, errors can be detected on thenoisy |CCZ

states.

To be more precise about the error correction properties of a circuit,as an example, take the following definitions:

Definition 1: Given two Boolean functions ƒ and g that can be expressedas cubic polynomials, it can be said they are Clifford-equivalent ƒ˜g ifand only if there exists a Boolean function q expressible as a quadraticfunction, such that ƒ(x)=g(x)+q(x) for all x.

If ƒ˜g, then clearly they also have the same cubic rank, and theassociated unitaries have the same minimal CCZ count.

Definition 2: Given a sequence of ζ generalized-CCZ gates described bythe set of column vectors, {J₁ ^(j),J₂ ^(j),J₃ ^(j)}_(j=1, . . . , ζ) asused in the equation above in the Synthesis discussion a set of matricesJ_(j) is defined each with three columns as follows:

$J^{j} = {( {J_{1}^{j},J_{2}^{j},J_{3}^{j}} ) = ( \frac{L^{j}}{C^{j}} )}$If the last m qubits are considered as check qubits, then the matricesare partitioned into C^(j) (the bottom m rows) and L^(j) as shown.

It is noted that C=(C¹, C², C³, . . . , C^(ζ)) and L=(L¹, L², . . . ,L^(ζ)) playa role analogous to X-check and logical X operator matricesof quantum code. Also error notation for the error patterns on theinitial magic states is needed.

Definition 3: Given a |CCZ

^(⊗ζ) magic state, it is said that it has error pattern e=(e₁, e₂, e₃)∈

₂ error if it is in the state E|CCZ

=(Z^(e) ¹ ⊗Z^(e) ² ⊗Z^(e) ³ )|CCZ

Given a sequence of generalized-CCZ gates, the notation used ise^(j)=(e₁ ^(j),e₂ ^(j),e₃ ^(j)) to denote the error for the jth |CCZ

state, so that the joint state is:E|CCZ

^(⊗k)=_(j=1) ^(ζ)⊗(Z ^(e) ¹ ^(j) ⊗Z ^(e) ² ^(j) ⊗Z ^(e) ³ ^(j) )|CCZ

It is said that an error has w fault-locations if e^(j) is non-zero forw of the |CCZ

states.

The distinction between notion used above with regard to weight and theusual Hamming weight of the concatenated string (e¹, . . . , e^(ζ)) isimportant because many methods of preparing a noisy |CCZ

state will lead to errors such as Z⊗Z⊗

that have a comparable probability to a single qubit error Z⊗Z⊗

. Indeed, we will typically be interested in knowing how many |CCZ

states are affected by an arbitrary error, though it is assumed errorsare uncorrelated between different |CCZ

states. Observations about error propagation in FIG. 25 can now beformalized as follows:

Given unitary U realized by a sequence of generalized CCZ gatesrepresented by matrices as in Def 2 with magic states suffering Paulierror (e¹, . . . , e^(ζ)). Then the resulting unitary on the targetqubits is UZ[w], where Z[w]=_(j=1) ^(n)⊗Z^(w) ^(n) and the vector w∈

₂ ^(n) is

$w = {\sum\limits_{j}{J^{j}e^{j}}}$Identifying the last m qubits as check qubits, w can be partitioned intotwo parts as follows:

${u = {\sum\limits_{j}{L^{j}e^{j}}}}{v = {\sum\limits_{j}{C^{j}e^{j}}}}$

Now knowing how errors propagate generally, this knowledge can beapplied to a specific protocol, such as distillation of 2 low-error ratelogical Toffoli gates from 8 noisy Toffoli magic states, or adistillation of 1 low-error rate logical Toffoli gate from 2 noisyToffoli magic states.

Consider a unitary U realized by a sequence of generalized CCZ gatesrepresented by matrices as in Def 2 and with the last m qubitsidentified as check qubits and U=U_(C)(U_(L)⊗

_(m)) where U_(C) is Clifford and

_(m) acts on the check qubits. Consider the following protocol:

-   -   1.) Prepare all qubits in the state |+        ;    -   2.) Perform the Clifford inverse U_(C) ⁻¹ and any Clifford        corrections from gate injection;    -   3.) Measure the last m qubits in the X basis.

Then X basis measurements in step 4 will yield +1 outcomes provided themagic state error pattern satisfies:

$( {0,0,\ldots\mspace{14mu},0} )^{T} = {\sum\limits_{j}{C^{j}e^{j}}}$The protocol outputs the magic state Z[u]U_(L)|+

^(⊗n) with error Z[u] that is trivial whenever

$( {0,0,\ldots\mspace{14mu},0} )^{T} = {\sum\limits_{j}{L^{j}e^{j}}}$Example Protocols

Consider the 2CCZ→1CCZ protocol with J^(j) matrices:

$( \frac{\begin{matrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{matrix}}{\begin{matrix}1 & 0 & 0\end{matrix}} ),( \frac{\begin{matrix}0 & 0 & 0 \\1 & 1 & 0 \\0 & 0 & 1\end{matrix}}{\begin{matrix}1 & 0 & 0\end{matrix}} )$It is straightforward to verify that the corresponding cubic polynomialis:ƒ(x)=x ₂ x ₃ x ₄ +x ₂ ² x ₃ =x ₂ x ₃ x ₄ +x ₂ ² x ₃ ˜x ₁ x ₂ x ₃

So the circuit realizes U=CZ_(2,3)(CCZ_(1,2,3))⊗

₄, which is a single CCZ gate and (up to a Clifford) it acts triviallyon the check qubit. There is only a single check qubit v₁=e₁ ¹+e₁ ².Therefore, it detects any error pattern where 1=e1¹+e₁ ², which includesa single (Z⊗

⊗

) error on either input magic state. However, it fails to detect othersingle fault error patterns such as (

⊗Z⊗

) on one magic state.

Now consider the 8CCZ→2CCZ protocol that detects an arbitrary error on asingle input CCZ state. A possible circuit 2602 implementation of thisprotocol is illustrated in FIG. 26. This protocol uses 3 check qubitsand the associated Jj matrices are shown in FIG. 26, such as matrix 2608corresponding to a first CCZ, matrix 2610 corresponding to a second CCZ,matrix 2612 corresponding to a third CZZ, and matrix 2614 correspondingto an eighth CCZ. Note that there would be eight total matrices with onecorresponding to each of the eight CCZ's. However, for ease ofillustration only matrices for CCZs 1-3 and 8 are shown. Computing thecubic polynomial, yields:ƒ(x)=x ₁ x ₂ x ₃ +x ₄ x ₅ x ₆which represents two CCZ gates and has trivial action on the checkqubits. Notice, there is no quadratic component to this polynomial, sono inverse Clifford is required. Regarding the error detectioncapabilities, notice that every check matrix is the identity and so thethree bit error syndrome is v=Σ_(j)e^(j). Given a fault on a single CCZstate, one of the e^(j) vectors will be non-zero and so v will benon-zero and the error is detected. In contrast, if two CCZ states havean identical error pattern, so e^(j)=e^(j′)≠0, then the syndromes willcancel and this will be an undetected error. However, not all two faulterrors go undetected. If magic states j and j′ suffer faults, bute^(j)≠e^(j′), then this two fault pattern will be detected. Theintuition for why the above matrices have the desired property isrelated to the fact that the matrices are built using a subset of thecodewords from 3 copies of a Reed-Muller code.

Consider an error model where a single noisy magic state has errorpattern e^(j) with probability

(e^(j)) We will use the convention

(0,0,0)=1−∈. The success probability is:

$P_{suc} = {\sum\limits_{0 = {\Sigma_{j}e^{j}}}{\prod\limits_{j}{{\mathbb{P}}( e^{j} )}}}$where the sum is over all configurations with trivial syndrome. Todetermine the fidelity of the output magic state, we should sum over allconfigurations with trivial syndrome and no logical damage on the state.To leading order this is dominated by the “no error” case, and indeedthis gives a firm lower bound on the fidelity, soƒ≥(1−∈)⁸ /P _(suc)Now considering the depolarizing error distribution:

$(e) = \{ \begin{matrix}{{1 -} \in} & {e = ( {0,0,0} )^{T}} \\{\in {/7}} & {e \neq ( {0,0,0} )^{T}}\end{matrix} $

Then the leading order contributions to the success probability can becounted as follows. The zero faults contribution adds to the successprobability. We do not count any single fault events since they are alldetected. Of the two fault events, we need a pair (j,j′) of magic states(of which there are 8 choose 2=28 combinations) to suffer the samenon-trivial error pattern e^(j), of which there are 7 types of e^(j)≠0.This means there are 196 undetected two fault error patterns, whichcontributes 196(∈/7)²(1∈)⁶=(196/49) ∈²(1−∈)⁶ to the success probability.However, not all of the undetected two fault error patterns lead to alogical failure, with a contribution of (184/49)∈²(1−∈)⁶ to undetectedlogical failures. This leads to the approximate results of:

${P_{suc} \approx {1 - 8}} \in \in_{out}{\approx ( \frac{184}{49} )} \in^{2}{\approx {{3.7}55}} \in^{2}$

Note that the constant factor 3.755 in front of ∈² is quite small for adistillation protocol. This is because this protocol detects the vastmajority of all two fault events.

In some embodiments, the above protocol may be generalized to 3k+2CCZ→kCCZ.

Example Implementation of Lattice Surgery

FIG. 27 illustrates an example implementation of the above describedprotocol using lattice surgery. Throughout, we refer to the input magicstate error rate as E and the output error rate is simply∈_(target)˜0(∈²). As noted earlier, the generalized CCZ gates can beinjected using only multi-Pauli measurements. For many error correctioncodes, such as topological codes and repetition codes, lattice surgeryprovides a natural way to measure multi-qubit Pauli operators. Thefollowing examples are concerning using thin surface codes withasymmetric distance for bit-flip and phase-flip noise. When there is anasymmetry we use the convention that the bit-flip distance is smaller.This also includes the repetition code as the limit where the bit-flipdistance is one.

The lattice surgery approach dedicates some ancilla qubits to act ascommunication routes between logical qubits. When performing amulti-patch measurement, these qubits are temporarily brought into anerror correction code for d_(m) of error correction. The value of d_(m)must be sufficiently large that the probability of an error during themulti-patch measurement is small enough. The larger d_(m), the moreprotection one has against measurement errors. However, an error duringmeasurement is equivalent to a single-qubit Pauli error on the magicstate. Therefore, d_(m) has to be sufficiently large that theprobability of measurement error is small than 0(∈). However, themeasurement error probability does not have to be smaller than theintended infidelity of the output magic state. However, the logicalqubits labelled 1 through 6, need to be encoded in a code protectingwith distance ok for bit-flips and d_(z) for phase-flips, where theseare sufficiently large that logical error rates are lower than O(∈²).

The logical qubits labelled 7 through 9 are the check qubits for theprotocol and are encoded in a code with distance d_(x) for bit-flips andd′_(z) for phase-flips. If there is a Z logical error on a check qubitat any point, this can be commuted to the end of the circuit and will bedetected provided it is the only fault. Therefore, we can setd′_(z)<d_(z), requiring only that d′_(z) is sufficiently large that a Zlogical occurring is less likely than O(∈). In the surface code, thespace/qubit cost is 2 d_(z)d_(x), so the total space cost for qubits 1through 9 and the routing ancilla space is:N ₁=14d _(x)(2d _(x)+2d _(z) +d′ _(z))

In addition, there is a space cost N₀ for the L₀ blocks responsible forpreparing the input Toffoli or CCZ states. We will need 8 such CCZstates, though in FIG. 27 the injection process is divided into twobatches of 4 CCZ states. Therefore, we need at least 4 L0 blocks, butdue to the finite success probability pf of each L0 block, someredundancy is needed to ensure we succeed with high probability(otherwise there will be a slight time delay). Given a factor Rredundancy, so we use 4R copies of the L0 blocks, the probability of allfailing is approximately 4p_(ƒ) ^(R). The size of the L0 blocks willdepends on the underlying protocol used, which in the case of the bottomprotocol is 3 d′_(z). If a factor R redundancy is require then the totalL0 space requirement is:N ₀=3Rd′ _(z)

In FIG. 27 a layout is shown where R=3. Note that if 2Rd′_(z)=7 d_(x),as in the FIG. 27, then the L0 blocks neatly line up with the ancillarouting region. If all the L0 blocks cannot fit adjacent to the routingregion and a different placement (such as having two columns of L0blocks) will need to be used.

The time cost of the whole distillation protocol is 10 d_(m) codecycles. Most of this cost is due to multi-patch measurements. Recallthat in FIG. 27 the injection process is divided into two batches of 4CCZ states. In each batch, there are several injection eventsinterspersed with each other, which is possible because all the gatesinvolved commute. Note also that the protocol uses multi-patchmeasurements and single-qubit measurements, but the single qubitmeasurements can be realized in 1 code-cycle and so are a negligiblecost. Assuming a surface code architecture where each code cycle takes4tCNOT where tCNOT is the CNOT gate time, gives a total 40 d_(m) tCNOTtime cost.

FIG. 28 illustrates a process for distilling low-error rate logicalToffoli gates from a plurality of noisy Toffoli magic states/Toffoligates, according to some embodiments.

At block 2802 a plurality of Toffoli magic states/noisy Toffoli gatesare prepared using a bottom-up approach or other suitable approach. Atblock 2804 a low-error rate logical Toffoli gate is distilled from aplurality of the Toffoli magic states/Toffoli gates prepared at block2802. At block 2806 a check qubit is measured to check for errors,wherein the check qubit is associated with the distilled low-error ratelogical Toffoli gate. At block 2808 a low-error rate logical Toffoligate operation is performed using the distilled low-error rate logicalToffoli gate in response to a verifying the check qubit does notindicate an error.

FIG. 29A illustrates a process of distilling two low-error rate logicalToffoli gates from eight noisy Toffoli magic states/Toffoli gates,according to some embodiments.

At block 2902 8 noisy Toffoli magic states/Toffoli gates are selected tobe used in a distillation of a low-error rate logical Toffoli gate. Atblock 2904 lattice surgery is performed to distil the one low-error ratelogical Toffoli gate from the 8 noisy Toffoli magic states/Toffoligates. At block 2906 a logical Toffoli gate operation is performed usingthe distilled low-error rate logical Toffoli gate, wherein a probabilityof error is quadratically suppressed for the low-error rate logicalToffoli gate as compared to the error rates of the 8 noisy Toffoli magicstates/Toffoli gates.

FIG. 29B illustrates a process of distilling a low-error rate logicalToffoli gate from two noisy Toffoli magic states/Toffoli gates,according to some embodiments.

At block 2952, two noisy Toffoli magic states/Toffoli gates are selectedto be used in a distillation of a low-error rate logical Toffoli gate.At block 2954 lattice surgery is performed to distil the one low-errorrate logical Toffoli gate from the 2 noisy Toffoli magic states/Toffoligates. At block 2956, a logical Toffoli gate operation is performedusing the distilled low-error rate logical Toffoli gate, wherein aprobability of very biased noise is quadratically suppressed for thelow-error rate logical Toffoli gate as compared to the very biased noiseof the 2 noisy Toffoli magic states/Toffoli gates.

FIG. 30 illustrates an example method of performing lattice surgery todistill a low-error rate logical Toffoli gate from a plurality of noisyToffoli magic states/Toffoli gates, according to some embodiments.

At block 3002 multi-qubit Pauli operator measurements are performedduring lattice surgery used to distill a low-error rate logical Toffoligate from noisy Toffoli magic states/Toffoli gates, wherein for eachJ_(k) with k=1, 2, 3, . . . the following steps are performed. Forexample, at block 3004, for each k value, a measurement ofZ_(k)·⊗Z[J_(k)] is measured where Z_(k) denotes Pauli Z acting on thek^(th) qubit of the magic state and Z[J_(k)] is a string of Paulioperators acting on the algorithmic qubits indexed by the binary vectorJ_(k). Also at block 3006, for each k, measure X on the k^(th) qubit ofthe magic state. At block 3008 for each “−1” outcome measured in step3006, update the Clifford correction frame by Z[J_(k)]. Then at block3010 using the measurement outcome from step 3004, update the Cliffordcorrection frame by the correction given in the figure.

High Fidelity Measurements

In some embodiments, low measurement error and/or faster errorcorrection can be achieved by using an additional readout mode that isinterrogated as the next error correction cycle proceeds. For example,circuit 3100 shown in FIG. 31 includes a readout qubit that enablesmeasurements 3106 to be performed for a first round of error correctiongates 3106 while (e.g. concurrently) a second round of error correctiongates 3104 are being performed.

Note that while some of the examples included herein are for hybridacoustic-electrical qubits and the architecture described in FIGS. 1-30,in some embodiments such measurement techniques could be applied inother architectures.

Consider fault tolerant operation of a quantum computer where properties(like stabilizers) of data qubits are repeatedly measured. In a givencycle of the error correction this often involves two steps. First gatesact between the data qubits and an ancilla qubit and then the ancillaqubit is measured. Subsequent to the measurement of the ancilla qubitanother error correction cycle can proceed.

In some embodiments, faster error correction cycles and lowermeasurement error can be achieve by swapping an ancilla (that wouldnormally be interrogated directly) to an additional readout qubit (couldbe some other gate that achieves same purpose as SWAP like iSWAP,decomposition of SWAP into CNOTS etc. Then perform readout on thereadout qubit while the rest of the error correction proceeds.

Such an approach not only reduces error correction cycle time, but alsoreduces idling errors on the data qubits. This is because the dataqubits only idle during the time of the swap is typically shorterduration that was is required to perform the measurements. Also, becauseidling is not a concern when measurements are performed on a readoutqubit, more repeated measurement may be taken, which also increasesmeasurement fidelity. For example, the full error correction cycle timemay be used to collect as many measurements as permitted to increasemeasurement fidelity or perform a single measurement with a longintegration time for the time of the next cycle.

For example, in a traditional surface code architecture with transmonsmeasurement is often much slower than the gates. The error correctioncycle time can be sped up by using this scheme. Additionally dependingon the details, one may have more time to drive/integrate allowing forhigher fidelity measurement without hurting the threshold because oflarge idling errors.

In some embodiments, the additional readout mode may be a bosonic mode.In such embodiments, for the measurement of the readout mode repeatedindividual parity measurements are performed which are then majorityvoted to determine the final outcome. Being able to take more of therepeated measurements increases the fidelity of the final outcome.

FIG. 32 illustrates a more specific example, wherein deflation isfurther added. Following the CNOT gates to entangle the ancilla qubit3204 with the data qubits 3202 the ancilla qubit is deflated. Deflationinvolves decreasing the steady state a for the dissipatively stabilizedancilla qubit from an initial |α_(initial)| to some |α_(final)|. Thedeflation provides protection from single photon loss events which occurat a rate proportional to the average number of bosons in the readoutmode. Once the mode has been deflated a SWAP 3212 is performed whichtransfers the excitation from the ancilla qubit 3204 to the bosonicreadout mode 3206 (which may be a phononic mode). To achieve highfidelity readout, repeated QND parity measurements of the bosonicreadout mode 3206 are employed. Each individual parity measurement isachieved by dispersively coupling the readout mode to a transmon qubit3208.

In some embodiments, during a parity measurement of a bosonic mode theaim is to determine whether there is an even or odd number of photons ina resonator. A single photon loss even during the process of ameasurement will change the parity potentially resulting in an incorrectreadout. For dissipatively stabilized systems a simple way to improvethe measurement fidelity is to perform a deflation operation 3214 beforethe measurement.

In the specific case of a system stabilized by two photon dissipationthis involves taking the dissipator D[a²−α_(initial) ²] toD[a²−α_(final) ²], wherein |α_(final)|<|α_(initial)|. This is done byvarying α(t) from the initial to final value. In most cases sufficientabrupt change is acceptable since there is no need to maintain phasecoherence between the even and odd parity states.

As is clear in the case without the deflation there is a significantdegradation in the infidelities as average photon number (α²) isincreased because the measurements are more sensitive to single photonloss which changes parity. With the deflation added this problem iscorrected.

As an example, FIG. 33 illustrates a parity measurement 3302 being takensubsequent to deflation.

In some embodiments, where a is the qubit mode and b is another modeused for readout, deflation can follow the following procedure:

-   -   1.) Deflate the qubit to α=0 mapping the + cat sate to |0        and the − cat sate to |1        .    -   2.) Evolve under a Hamiltonian H=ig(b^(℄)−b)a^(†)a and measure        (homodyne/heterodyne) the b mode to determine whether the qubit        was in a + or − cat state. If the qubit was in the − cat state        then there is a drive on the b mode implemented by the        Hamiltonian whereas if the qubit was in the + cat state there is        no drive on the b mode. Hamiltonians of this form can be derived        resonantly and non-resonantly from a three wave mixing        Hamiltonian of the form:        H˜∈(t)(φ_(a)(a+a ^(†))+φ_(b)(b+b ^(†)))³

In some embodiments, other Hamiltonians may be used, such asH=g(b^(†)+b)a^(†)a or H=ig(b^(†)−b)a^(†)a.

In some embodiments, bosonic modes may be readout in +|α

basis using a three or higher wave mixing Hamiltonian. In someembodiments a procedure for such readouts may comprise evolving under aHamiltonian H=g (a^(†)b+b^(†)a) and measuring (homodyne/heterodyne) theb mode to measure the bosonic mode in ±|α

basis. Hamiltonians of this form can be derived resonantly andnon-resonantly from a three wave mixing Hamiltonian of the form:H˜∈(t)(φ_(a)(a+a ^(†))+φ_(b)(b+ab ^(†)))³

In some embodiments, other Hamiltonians may be used, such asH=g(a+a^(†))(b+b^(†)) or H=g(a+a^(†)) (b−b^(†)), etc.

FIG. 34 is a process flow diagram illustrating using a switch operatorto excite a readout qubit such that a subsequent round of errorcorrection gates can be applied in parallel with performing measurementsof the readout qubit, according to some embodiments.

At block 3402, a set of error correction gates is applied between dataqubits storing quantum information and an ancilla qubit. At block 3404,a swap is performed between the ancilla qubit and a readout qubit. Atblock 3406, one or more measurements are performed on the readout qubit.While this is taking place or without waiting for the measurements atblock 3406 to complete, at block 3408 another set of error correctiongates are applied between data qubits storing the quantum informationand the ancilla qubit. At block 3410, another swap is performed betweenthe ancilla qubit and the readout qubit, subsequent to the measurementat block 3406 completing. And, at block 412 one or more measurement areperformed on the readout qubit. Note that this process can be repeatedfor multiple additional rounds of error correction.

FIG. 35 is a process flow diagram illustrating a process for usingdeflation or evolution using a three or higher wave mixing Hamiltonianto perform measurements of an ancilla qubit without requiring a transmonqubit, according to some embodiments.

As an example, one or more data qubits storing quantum information maybe entangled with an ancilla qubit. At block 3502 a qubit, such as theancilla qubit, is deflated prior to performing a readout of the qubit,such that phonons or photons are dissipated from the qubit while ameasurement observable of the qubit is preserved. Then at block 3504, areadout of the measurement observable of the deflated qubit isperformed.

FIG. 36A is a process flow diagram illustrating a process for deflatinga cat qubit and measuring a b mode of the deflated cat qubit todetermine information about a first mode of the deflated cat qubit,according to some embodiments.

At block 3602 cat qubit is deflated such that phonons or photons aredissipated from the cat qubit. For example, this may be achieved byadjusting a steady state dissipation rate, for example as may be drivenby an ATS. At block 3604, the cat qubit is evolved under Hamiltonianthat couples a number of excitations of the cat qubit to a second mode(b mode) of the cat qubit. Then, at block 3606, the second mode (e.g. bmode) of the cat qubit is measured to determine information about thefirst mode (e.g. a mode) of the cat qubit.

FIG. 36B is a process flow diagram illustrating another process fordeflating a qubit and measuring a “b” mode of the deflated cat qubit todetermine information about a first mode of the deflated cat qubit,according to some embodiments.

At block 3652 deflation is performed in a system wherein an “a” mode isa qubit mode and a “b” mode is a readout mode. The deflation includesdeflating a qubit to α=0 such that the + cat sate is mapped to |0

and the − cat sate is mapped to |1

. At block 3654, the system is evolved under a Hamiltonian derived froma three wave or higher mixing Hamiltonian. For example, a HamiltonianH=ig(b^(†)−b)a^(†)a. At block 3656 measurements of the “b” mode areperformed to determine whether the qubit is in the + or − cat state. Forexample, (homodyne/heterodyne) measurements of the b mode are performedto determine whether the qubit was in a + or − cat state. If the qubitwas in the − cat state then there is a drive on the “b” mode implementedby the Hamiltonian whereas if the qubit was in the + cat state there isno drive on the “b” mode.

FIG. 37 is a process flow diagram illustrating a process for evolving acat qubit via three wave or higher mixing Hamiltonian and performing ahomodyne, heterodyne, or photo detection of the evolved cat qubit tomeasure a measure property of another bosonic mode of the cat qubit,according to some embodiments.

At block 3702, a cat qubit is evolved under a Hamiltonian that couples aphase of the cat qubit to a measurable property of another bosonic modeof the cat qubit, wherein the Hamiltonian is selected from a three waveor higher mixing Hamiltonian. At block 3704, a homodyne, heterodyne, orphoto detection of the other bosonic mode is performed to determine thephase of the cat qubit.

Simulation of Cat Qubits Using a Shifted Fock Basis

A Fock basis is an algebraic construction used to construct quantumstate space for a variable or unknown number of identical particlesbased on a single particle in Hilbert space. For example, a Fock basiscould be used to simulate a cavity or the behavior of a phononicresonator using an n-dimensional ladder of states. For example, Fockbasis may be used to simulate photon number states, wherein a base staterepresents a vacuum condition without any photons present. However, byshifting the Fock basis, the Hilbert space can be truncated to include afinite (as opposed to infinite) number of photon number states. Thus,simulations can be simplified such that the truncated Hilbert space issimulated as opposed to the infinite Hilbert space, which cannot beeffectively simulated. As an example, a shifted Fock basis simulationmay replace a vacuum state with one or more coherent states. Forexample, a shift operator may be applied to the vacuum state conditionsuch that the lowest shifted Fock states correspond to the lowestoperators for the lowest states of a cat qubit.

For example, simulating a large cat qubit (with large |α²|>>1) using atraditional (e.g. non-shifted) Fock basis may be ineffective due to thelarge (or even infinite) number of states that would need to besimulated. Instead, in some embodiments, the simulation may be performedusing a shifted Fock basis, which can be used to describe large catstates in a more compact way than is the case for a usual Fock basis.More specifically, the annihilation operator a may be constructed in ashifted Fock basis.

Recall that a cat state is composed of two coherent state components |±α

which can be understood as displaced vacuum states {circumflex over(D)}(±α)|{circumflex over (n)}=0

. In the shifted Fock basis, 2 d displaced Fock states {circumflex over(D)}(±α)|{circumflex over (n)}=n

are used as basis states where n∈{0, . . . , d−1}. Note that whiledisplaced Fock states in each ±α branch are orthonormalized, displacedFock states in different branches are not necessarily orthogonal to eachother. Thus the displaced Fock states need to be orthonormalized.

The non-orthonormalized basis states may be defined as follows:

$ {\phi_{n}, \pm} \rangle \equiv {{\frac{1}{\sqrt{2}}\lbrack {{\hat{D}(\alpha)} \pm {( {- 1} )^{n}{\hat{D}( {- \alpha} )}}} \rbrack} {\overset{\hat{}}{n} = n} \rangle}$where |ϕ_(n), +

and |ϕ_(n), −

have even and odd excitation number parity, respectively. Note that thenon-orthonormalized states are grouped into the even and odd branchesinstead of the ±α branches. As a result, in the ground state manifold(n=0), the normalized basis states |ϕ₀, ±

are equivalent to the complementary basis states of the cat qubit |±

, not the computational basis states |0/1

. For example:

$ {{{{  \pm \rangle \propto}}\phi_{0}}, \pm} \rangle = {\frac{1}{\sqrt{2}}( { \alpha \rangle \pm  {- \alpha} \rangle} )}$

The even/odd branching convention is used so that any two basis statesin different branches are orthogonal to each other and hence theorthonormalization can be done separately in each parity sector. Notethat:Φ_(m,n) ^(±)≡

ϕ_(m,±)|ϕ_(n,±)

=δ_(m,n)(−1)^(m) D _(m,n)(2α),where D_(m,n)(α)≡

{circumflex over (n)}=m|{circumflex over (D)}(α)|{circumflex over (n)}=n

are the matrix elements of the displacement operator {circumflex over(D)}(α) in the usual Fock basis:

${D_{m,n}(\alpha)} = {e^{- \frac{{\alpha }^{2}}{2}}\sqrt{\frac{{\min( {m,n} )}!}{{\max( {m,n} )}!}}{L_{\min{({m,n})}}^{({{m - n}})}( {\alpha }^{2} )} \times \{ \begin{matrix}\alpha^{m - n} & {m \geq n} \\( {- \alpha^{*}} )^{n - m} & {m < n}\end{matrix} }$

Here, L_(n) ^((α))(x) is the generalized Laguerre polynomial. Since|D_(m,n)(2α)|=O(|α|^(m+n)e^(−2|α|) ² ), D_(m,n)(2α) is negligible ifm+n«|α|². In this regime, the basis states |ϕ_(n), ±

are almost orthonormal. For the purpose of estimating the phase-flip (orZ) error rates within a small multiplicative error, it is oftenpermissible to neglect the non-orthogonality of the states |ϕ_(n), ±

. However, this is generally not the case if the Z error rates are to beevaluated with a very high precision or if it is desired to estimate thebit-flip (or X) error rates. In these cases, taking into account thenon-orthogonality of the states |ϕ_(n), ±

may be necessary.

In such embodiments, the basis states |ϕ_(n), ±

are orthonormalized by applying the Gram-Schmidt orthonormalizationprocedure. More specifically, given the non-orthonormalized basis states|ϕ_(n,±)

, d orthonormalized basis states are constructed in each parity sectorstarting from the ground state |ϕ_(0,±)

:

$ \psi_{n, \pm} \rangle = {\sum\limits_{m = 0}^{d - 1}{c_{m,n}^{\pm} \phi_{m, \pm} \rangle}}$

The coefficients of c_(m,n) ^(±)(0≤m, n≤d−1) are determined inductively.In the base case (k=0),

${c_{0,0}^{\pm} = \frac{1}{\sqrt{\Phi_{0,0}^{\pm}}}},{c_{m,0}^{\pm} = {{0\mspace{14mu}{for}\mspace{14mu}{all}\mspace{14mu} 1} \leq m \leq {d - 1}}},$and thus the logical |±> states of the cat qubit are given by:

${{{{  \pm \rangle \equiv  \psi_{0, \pm} \rangle} = \frac{1}{\sqrt{\Phi_{0,0}^{\pm}}}}} \phi_{0, \pm} \rangle} = \frac{ \alpha \rangle \pm  {- \alpha} \rangle}{\sqrt{2( {1 \pm e^{{- 2}|\alpha|^{2}}} )}}$

In general, the case with 1≤k≤d−1, we are given with c_(m,n) ^(±) forall 0≤m≤d−1 and 0≤n≤k−1. Thus, at this point, the first k columns ofc^(±) are known. Let c_(:0:k−1) ^(±) be the d×k matrix which is obtainedby taking the first k columns of the matrix c^(±). Given c_(:0:k−1)^(±), we assign the k+1 column of c^(±) as follows:

$c_{m,k}^{\pm} = {- \frac{( {{c_{:{,{0:{k - 1}}}}^{\pm}( c_{:{,{0:{k - 1}}}}^{\pm} )}^{\dagger}\Phi^{\pm}} )_{m,k}}{\sqrt{\Phi_{k,k}^{\pm} - ( {{( \Phi^{\pm} )\dagger\;{c_{:{,{0:{k - 1}}}}^{\pm}( c_{:{,{0:{k - 1}}}}^{\pm} )}^{\dagger}\Phi} \pm} )_{k,k}}}}$for 0≤m≤k−1,

$c_{k,k}^{\pm} = \frac{1}{\sqrt{\Phi_{k,k}^{\pm} - ( {( \Phi^{\pm} )\dagger\;{c_{:{,{0:{k - 1}}}}^{\pm}( c_{:{,{0:{k - 1}}}}^{\pm} )}^{\dagger}\Phi^{\pm}} )_{k,k}}}$and c_(m,k) ^(±)=0 for all m>k.

Having constructed the 2 d orthonormalized shifted Fock basis states|ψ_(n,±)

the matrix elements for an operator Ô (e.g. Ô=â) in the orthonormalizedbasis need to be determined. To do this, let |ϕ_(n)

=|ϕ_(n)+

and |ϕ_(n+d)

=|ϕ_(n,−)

for ∈{0, . . . , d−1} and also define |ψ_(n)

and |ψ_(n+d)

similarly. Suppose the operator Ô transforms the non-orthonormalizedbasis states |ϕ_(n)

as follows:

${\overset{\hat{}}{O} \phi_{n} \rangle} = {\sum\limits_{m = 0}^{{2d} - 1}{O_{m,n} \phi_{m} \rangle}}$

O_(m,n) are the matrix elements of the operator Ô in thenon-orthonormalized basis |ϕ_(n)

. Then, in the orthonormalized basis, the matrix elements of theoperator Ô are given by:O′ _(m,n)≡

ψ_(m) |Ô|ψ _(n)

=c ^(†) ΦOc)_(m,n)where Φ and c are 2 d×2 d matrices which are defined as:

${\Phi = \begin{bmatrix}\Phi^{+} & 0 \\0 & \Phi^{-}\end{bmatrix}},{c = \begin{bmatrix}c^{+} & 0 \\0 & c^{-}\end{bmatrix}}$

The matrix elements of the d×d matrices Φ^(±) and c^(±) are given above.

Consider the annihilation operator Ô=â and note that it transforms thenon-orthonormalized basis states |ϕ_(n,±)

as follows:â|ϕ _(n,±)

=√{square root over (n)}|ϕ_(n−1,∓)

+α|ϕ_(n,∓)

Here, the ±parity is flipped to the ∓ parity. Thus, in thenon-orthonormalized basis, the matrix elements of the annihilationoperator are given by:

$\begin{bmatrix}0 & {\overset{\hat{}}{b} + \alpha} \\{\overset{\hat{}}{b} + \alpha} & 0\end{bmatrix} = {\overset{\hat{}}{X} \otimes ( {\overset{\hat{}}{b} + \alpha} )}$

where {circumflex over (X)} is the Pauli X operator and {circumflex over(b)} is the truncated annihilation operator of size d×d. Then, thematrix elements of the annihilation operator in the orthonormalizedbasis |ψ_(n,±)

can be obtained via the transformation given above with regard toO′_(m,n).

Recall that |ψ_(n,±)

are complementary basis states. To find the matrix elements of anoperator in the computational basis states, the matrix may be conjugatedby the Hadamard operator Ĥ. Thus, in the orthonormalized computationalbasis, the annihilation operator is given by:

${{\hat{a}}_{SF} \equiv {{( {\hat{H} \otimes \hat{I}} ) \cdot c^{\dagger}}{\Phi( {\hat{X} \otimes ( {\hat{b} + \alpha} )} )}{c \cdot ( {\hat{H} \otimes \hat{I}} )}}}\overset{{\alpha }^{2} ⪢ d}{arrow}{\hat{Z} \otimes ( {\hat{b} + \alpha} )}$

Here the subscript SF indicates the action of the annihilation operatorin the shifted Fock basis. The approximate expression â_(SF)≅{circumflexover (Z)}⊗({circumflex over (b)}+α) is useful for analyzing the Z errorrates of large cat qubits (with |α|²>>1) in the perturbative regimewhere the cat qubit states may sometimes be excited to the first excitedstate manifold (n=1) but quickly decay back to the ground state manifold(n=0). Lastly, it is noted that the parity operatore^(i{circumflex over (π)}â†â) is exactly given by {circumflex over(X)}└Î in the shifted Fock basis because of the way the basis states aredefined, e.g., |ψ_(n,+)

(|ϕ_(n,−)

) has an even (odd) excitation number parity.

FIG. 38 is a process flow diagram illustrating a process of utilizing ashifted Fock basis to simulate a cat qubit (with |α|²>>1), according tosome embodiments.

At block 3802, non-orthonormalized basis states are defined as describedabove. At block 3804 the basis states are orthonormalized to construct 2d orthonormalized shifted Fock basis states as described above. At block3806 matrix elements are determined for an operator in theorthonormalized basis as described above.

Embodiments of the present disclosure can be described in view of thefollowing clauses:

-   Clause 1. A system, comprising:    -   a mechanical linear resonator; and    -   a control circuit coupled with the mechanical linear resonator,    -   wherein the control circuit is configured to stabilize an        arbitrary coherent state superposition (cat state) of the        mechanical linear resonator to store quantum information,        wherein to stabilize the arbitrary cat-state, the control        circuit is configured to:        -   excite phonons in the mechanical linear resonator by driving            a storage mode of the mechanical linear resonator; and        -   dissipate phonons from the mechanical linear resonator via            an open transmission line coupled to the control circuit            configured to absorb photons from a dump mode of the control            circuit.-   Clause 2. The system of clause 1, wherein the control circuit    comprises:    -   an asymmetrically-threaded superconducting quantum interference        device (ATS) coupled with the mechanical resonator.-   Clause 3. The system of clause 2, further comprising:    -   one or more additional mechanical linear resonators coupled to        the control circuit, wherein the control circuit is configured        to stabilize respective cat states of the mechanical resonator        and the one or more additional mechanical linear resonators via        the single ATS and the single open transmission line.-   Clause 4. The system of clause 3, wherein the storage modes of the    respective mechanical linear resonators are detuned, such that the    phonons supplied to the respective mechanical linear resonators are    supplied in an incoherent manner.-   Clause 5. The system of clause 4, wherein pumps of the respective    mechanical linear resonators are separated by a frequency bandwidth    greater than a two-phonon dissipation rate of the respective    mechanical linear resonators.-   Clause 6. The system of clause 5, wherein the control circuit    further comprises:    -   one or more microwave filters configured to filter out        correlated decay terms that if not filtered out cause        simultaneous phase flips of storage modes of two or more of the        mechanical linear resonators.-   Clause 7. The system of clause 2, wherein the control circuit    further comprises:    -   a high-impedance inductor used as part of the ATS coupled to the        mechanical linear resonator.-   Clause 8. The system of clause 7, wherein the high-impedance    inductor comprises:    -   a planar meander or double-spiral inductor;    -   a spiral inductor with one or more air bridges;    -   an array of Josephson junctions; or    -   a thin-film superconductor with a high kinetic inductance.-   Clause 9. The system of clause 2, wherein at least some of the    mechanical linear resonators comprise three or more terminals, the    system further comprising:    -   two or more additional asymmetrically-threaded superconducting        quantum interference devices (ATS),    -   wherein a given one of the mechanical linear resonators        comprising three or more terminals is coupled with three or more        ATSs via the respective three or more terminals.-   Clause 10. A method of stabilizing coherent state superpositions    (cat states) of a mechanical resonator, the method comprising:    -   exciting phonons in the mechanical resonator by driving a        storage mode of the mechanical resonator; and    -   dissipating phonons from the mechanical resonator via an open        transmission line coupled to the control circuit configured to        absorb photons from a dump mode of the control circuit.-   Clause 11. The method of clause 10, wherein the phonons are excited    in the mechanical resonator and dissipated from the mechanical    resonator in pairs comprising two phonons.-   Clause 12. The method of clause 11, wherein the excitation and    dissipation of the phonon pairs is induced via a non-linear    interaction between the storage mode of the mechanical resonator and    the dump mode of the control circuit, wherein a square of the    storage mode of the mechanical resonator is coupled to the dump mode    of the control circuit via a two-phonon coupling rate (g₂), and    wherein a decay rate at which photons are absorbed via the open    transmission line is approximately ten times or greater than the    coupling rate (g₂).-   Clause 13. The method of clause 11, wherein the control circuit    comprises an asymmetrically-threaded superconducting quantum    interference device (ATS) coupled with the mechanical resonator,    wherein the ATS is configured to cause the two-phonon pairs to be    excited in the mechanical resonator.-   Clause 14. The method of clause 13, further comprising:    -   causing phonons to be excited in one or more additional        mechanical resonator by driving respective storage modes of the        one or more additional mechanical resonators; and    -   dissipating phonons from the one or more additional mechanical        resonators via the open transmission line configured to absorb        the photons from the dump mode of the control circuit,    -   wherein the single ATS is used to cause the phonons to be        excited in the mechanical resonator and the one or more        additional mechanical resonators.-   Clause 15. The method of clause 14, wherein the storage modes of the    respective mechanical resonators are detuned.-   Clause 16. The method of clause 15, wherein the storage modes of the    respective mechanical resonators are separated by a frequency    bandwidth greater than a two-phonon dissipation rate of the dump    mode of the control circuit.-   Clause 17. The method of clause 16, further comprising:    -   filtering out, via one or more microwave filters, correlated        decay terms of storage modes of two or more of the mechanical        resonators.-   Clause 18. A method of stabilizing coherent state superpositions    (cat states) of a plurality of resonators storing quantum    information, the method comprising:    -   causing, via a single asymmetrically-threaded superconducting        quantum interference device (ATS), pairs of two phonons or pairs        of two photons to be excited in respective ones of the        respective resonators by driving respective storage modes of the        respective resonators; and    -   dissipating pairs of two photons from a dump mode of a control        circuit comprising the ATS, wherein the control circuit is        coupled with the respective resonators, and wherein an open        transmission line is coupled to the dump mode of the control        circuit.-   Clause 19. The method of clause 18, wherein the resonators are    mechanical resonators.-   Clause 20. The method of clause 18, wherein the resonators are    electromagnetic resonators.-   Clause 21. A method, comprising:    -   implementing a multi-qubit gate among control and target qubits        in a system comprising resonators and an asymmetrically-threaded        superconducting quantum interference device (ATS), wherein        implementing the multi-qubit gate comprises:        -   implementing a linear drive for a phononic mode of a cat            qubit for the gate, wherein the cat qubit is implemented via            one of the resonators of the system;        -   orchestrating Hamiltonian interactions, wherein the            Hamiltonian interactions comprise a compensating Hamiltonian            for the multi-qubit gate, and wherein the compensating            Hamiltonian includes a frequency shift of a target mode and            a control mode at the mechanical resonator being driven,            wherein the control mode and the target mode are coupled via            an optomechancical coupling.-   Clause 22. The method of clause 21, wherein a setting for the    multi-qubit gate comprises multiple ones of the resonators coupled    to the ATS, wherein the ATS is shared by the multiple ones of the    resonators.-   Clause 23. The method of clause 21, wherein the optomechanical    coupling is realized by off-resonantly driving the resonators and    the ATS.-   Clause 24. The method of 23, wherein said off-resonantly driving the    resonators and the ATS avoids frequency collisions.-   Clause 25. The method of clause 21, wherein the multi-qubit gate is    a CNOT gate.-   Clause 26. The method of clause 21, wherein the multi-qubit gate is    a Toffoli gate.-   Clause 27. The method of clause 21, wherein the resonators are    mechanical resonators.-   Clause 28. The method of clause 21, wherein the resonators are    electromagnetic resonators.-   Clause 29. A method for simulating a Toffoli gate encoded in    arbitrary Calderbank-Shor-Steane codes, the method comprising:    -   preparing computational basis states in a fault-tolerant manner        by applying a STOP algorithm to determine when syndrome        measurements of stabilizers of a repetition code for the        computational basis states can be stopped such that a        probability of faults for the computational basis states are        below a threshold level;    -   transversally applying a CNOT gate to the prepared computational        basis states to prepare a |ψ₁        state;    -   measuring a Clifford stabilizer g_(A) for the |ψ₁        state, and applying a logical Z correction if the measurement        outcome for the Clifford stabilizer g_(A) is −1, wherein        measuring the Clifford stabilizer g_(A) and applying the logical        Z correction based on a measurement outcome of the Clifford        stabilizer g_(A) prepares a state |ψ_(out)        ;    -   repeating the Clifford stabilizer g_(A) measurement for the |ψ₁        state a threshold number of times;    -   preparing a Toffoli magic state in response to determining the        Clifford stabilizer g_(A) measurements are trivial; and    -   applying a sequence of Clifford gates to a logical input state        ∥ψ        _(L) and the prepared Toffoli magic state to simulate the        logical Toffoli gate, wherein Clifford error corrections are        applied to the outputs of the sequence of Clifford gates applied        to the logical inputs.-   Clause 30. The method of clause 29, wherein applying the STOP    algorithm comprises:    -   tracking consecutive syndrome outcomes;    -   computing a minimum number of faults capable of causing a        tracked sequence of consecutive syndrome outcomes;    -   stopping the STOP algorithm if either of the following        conditions is met:        -   1) a same syndrome outcome is repeated a threshold number of            times in a row, wherein the threshold is equal to one plus a            difference between:            -   a code distance of one of the computational basis states                being prepared minus one wherein the result of the                subtraction is divided by two; and            -   a currently computed minimum number of faults capable of                causing the tracked sequence of consecutive syndrome                outcomes; or        -   2) the currently computed minimum number of faults capable            of causing the tracked sequence of consecutive syndromes is            equal to the code distance of the one of the computational            basis states being prepared minus one wherein the result of            the subtraction is divided by two, and wherein one            additional round of syndrome measurements is performed            subsequently; and    -   utilizing the repeated syndrome if condition 1 is met or        utilizing the syndrome for the subsequently performed syndrome        measurement if condition 2 is met, wherein the utilized syndrome        it utilized to error correct the one of the computational basis        states being prepared.-   Clause 31. The method of clause 30, wherein:    -   repeating the measurement of the Clifford stabilizer g_(A) for        the |ψ₁        state the threshold number of times comprises repeating the        measurement such that the Clifford stabilizer g_(A) is measured        a number of times equal to (d−1)/2, wherein d is a code distance        of the one of the fault tolerant computational basis states.-   Clause 32. The method of clause 31, wherein error detection is    performed between respective measurements of the Clifford stabilizer    g_(A).-   Clause 33. The method of clause 29, further comprising:    -   growing the Toffoli magic state from a first code distance to a        second code distance, wherein the STOP algorithm is used to        measure stabilizers and minimum weight perfect matching (MWPM)        is applied to a measured syndrome history generated from        measuring the stabilizers to correct for errors.-   Clause 34. A method, comprising:    -   measuring syndrome outcomes of an ancilla qubit for an arbitrary        Calderbank-Shor Steane code;    -   tracking consecutive ones of the measured syndrome outcomes;    -   computing a minimum number of faults capable of causing a        tracked sequence of consecutive syndrome outcomes;    -   stopping the measuring of the syndrome outcomes if either of the        following conditions is met:        -   1) a same syndrome outcome is repeated a threshold number of            times in a row, wherein the threshold is equal to one plus a            difference between:            -   a code distance of the arbitrary Calderbank-Shor-Steane                code minus one wherein the result of the subtraction is                divided by two; and            -   a currently computed minimum number of faults capable of                causing the tracked sequence of consecutive syndrome                outcomes; or        -   2) the currently computed minimum number of faults capable            of causing the tracked sequence of consecutive syndromes is            equal to the code distance minus one wherein the result of            the subtraction is divided by two, and wherein one            additional round of syndrome measurements is performed            subsequently; and    -   utilizing the repeated syndrome outcome if condition 1 is met or        utilizing the syndrome outcome for the subsequently performed        syndrome measurement if condition 2 is met, wherein the utilized        syndrome outcome is utilized to error correct the arbitrary        Calderbank-Shor-Steane code.-   Clause 35. The method of clause 34, wherein:    -   the arbitrary Calderbank-Shor-Steane code is a n-qubit        repetition code;    -   measuring the syndrome outcomes comprises measuring Z_(L) at the        ancilla for the n-qubit repetition code; and    -   performing the error correction for the n-qubit arbitrary        Calderbank-Shor-Steane code further comprises applying an XL        correction based on the measured Z_(L) at the ancilla for the        n-qubit repetition code,    -   wherein performing the error correction prepares computational        basis state to be used in implementing a Clifford gate.-   Clause 36. The method of clause 34, further comprising:    -   growing the arbitrary Calderbank-Shor-Steane code from a first        code distance to a second code distance, wherein a STOP        algorithm is used to measure stabilizers and minimum weight        perfect matching (MWPM) is applied to a measured syndrome        history generated from measuring the stabilizers to correct for        errors, wherein the STOP algorithm comprises said measuring        syndrome outcomes, said tracking consecutive ones of the        measured outcomes, said computing a minimum number of faults,        said stopping the measuring if condition 1 or condition 2 is        met, and said error correction.-   Clause 37. The method of clause 36, wherein said growing the    arbitrary Calderbank-Shor-Steane code from the first code distance    to the second code distance comprises:    -   performing lattice surgery to merge together two code blocks,        wherein the measuring comprises measuring a boundary operator        between the two code blocks being merged.-   Clause 38. The method of clause 34, further comprising:    -   preparing computational basis states in a fault tolerant manner        by applying a STOP algorithm to determine when syndrome        measurements of stabilizers of a repetition code for the        computational basis states can be stopped such that a        probability of faults for the computational basis states are        below a threshold level,    -   wherein:        -   the computational basis states are encoded using the            arbitrary Calderbank-Shor-Steane code; and        -   applying the STOP algorithm comprises performing said            measuring syndrome outcomes, said tracking consecutive ones            of the measured outcomes, said computing a minimum number of            faults, said stopping the measuring if condition 1 or            condition 2 is met, and said error correction.-   Clause 39. The method of clause 38, further comprising:    -   transversally applying a CNOT gate to the prepared computational        basis states to prepare a |ψ₁        state;    -   measuring a Clifford stabilizer g_(A) for the |ψ₁        state, and applying a logical Z correction if the measurement        outcome for the Clifford stabilizer g_(A) is −1, wherein        measuring the Clifford stabilizer g_(A) and applying the logical        Z correction based on a measurement outcome of the Clifford        stabilizer g_(A) prepares a state |ψ₁        ;    -   repeating the Clifford stabilizer g_(A) measurement for the |ψ₁        state a threshold number of times;    -   preparing a Toffoli magic state in response to determining the        Clifford stabilizer g_(A) measurements are trivial; and    -   applying a sequence of Clifford gates to the |ψ₁        state and the prepared Toffoli magic state to simulate a Toffoli        gate, wherein Clifford error corrections are applied to the        outputs of the sequence of Clifford gates applied to a logical        input.-   Clause 40. The method of clause 39, wherein:    -   repeating the measurement of the Clifford stabilizer g_(A) for        the |ψ₁        state the threshold number of times comprises repeating the        measurement such that the Clifford stabilizer g_(A) is measured        a number of times equal to (d−1)/2,        -   wherein d is a code distance of the one of the fault            tolerant computational basis states.-   Clause 41. The method of clause 40, wherein error detection is    performed between respective measurements of the Clifford stabilizer    g_(A).-   Clause 42. The method of clause 41, further comprising:    -   growing the Toffoli magic state from a first code distance to a        second code distance, wherein the STOP algorithm is used to        measure stabilizers and minimum weight perfect matching (MWPM)        is applied to a measured syndrome history generated from        measuring the stabilizers to correct for errors.-   Clause 43. The method of clause 34, wherein the arbitrary    Calderbank-Shor-Steane code and the ancilla are implemented using a    system comprising:    -   mechanical linear resonators; and    -   a control circuit coupled with the mechanical linear resonators,    -   wherein the control circuit is configured to stabilize an        arbitrary coherent state superposition (cat state) of the        mechanical linear resonators to store quantum information of the        Calderbank-Shor-Steane code, wherein to stabilize the arbitrary        cat-state, the control circuit is configured to:        -   excited phonons in the mechanical linear resonators by            driving respective storage modes of the mechanical linear            resonators; and        -   dissipate phonons from the mechanical linear resonators via            an open transmission line coupled to the control circuit            configured to absorb photons from a dump mode of the control            circuit.-   Clause 44. The method of clause 43, wherein the control circuit    comprises:    -   an asymmetrically-threaded superconducting quantum interference        device (ATS) coupled with the mechanical linear resonators.-   Clause 45. A system comprising:    -   mechanical resonators; and    -   a control circuit coupled with the mechanical resonators,        wherein the control circuit is configured to stabilize arbitrary        coherent state superpositions (cat states) of the mechanical        resonators to store quantum information; and    -   one or more computing devices storing program instructions, that        when executed cause the control circuit to perform:        -   measuring syndrome outcomes of an ancilla qubit for one or            more qubits storing the quantum information, wherein the            ancilla qubit and the one or more qubits storing the quantum            information are implemented via one or more of the            mechanical resonators;        -   tracking consecutive ones of the measured syndrome outcomes;        -   computing a minimum number of faults capable of causing a            tracked sequence of consecutive syndrome outcomes;        -   stopping the measuring of the syndrome outcomes if either of            the following conditions is met:            -   1) a same syndrome outcome is repeated a threshold                number of times in a row, wherein the threshold is equal                to one plus a difference between:                -   a code distance of the one or more qubits storing                    quantum information minus one wherein the result of                    the subtraction is divided by two; and                -   a currently computed minimum number of faults                    capable of causing the tracked sequence of                    consecutive syndrome outcomes; or            -   2) the currently computed minimum number of faults                capable of causing the tracked sequence of consecutive                syndromes is equal to the code distance minus one                wherein the result of the subtraction is divided by two,                and wherein one additional round of syndrome                measurements is performed subsequently; and    -   utilizing the repeated syndrome if condition 1 is met or        utilizing the syndrome outcome for the subsequently performed        syndrome measurement if condition 2 is met, wherein the utilized        syndrome outcome is utilized to error correct the stored quantum        information.-   Clause 46. The system of clause 45, wherein the one or more    computing devices are further configured to implement:    -   preparing computational basis states in a fault tolerant manner        by applying a STOP algorithm to the fault-tolerant computational        basis states to determine when syndrome measurements of        stabilizers of a repetition code for the computational basis        states can be stopped such that a probability of faults for the        computational basis states are below a threshold level,    -   wherein:        -   applying the STOP algorithm comprises performing said            measuring syndrome outcomes, said tracking consecutive ones            of the measure outcomes, said computing a minimum number of            faults, said stopping the measuring if condition 1 or            condition 2 is met, and said error correction.-   Clause 47. The system of clause 45, wherein the one or more    computing devices are further configured to implement:    -   transversally applying a CNOT gate to the prepared computational        basis states to prepare a |ψ₁        state;    -   measuring a Clifford stabilizer g_(A) for the |ψ₁        state, and applying a logical Z correction if the measurement        outcome for the Clifford stabilizer g_(A) is −1, wherein        measuring the Clifford stabilizer g_(A) and applying the logical        Z correction based on a measurement outcome of the Clifford        stabilizer g_(A) prepares a state |ψ₁        ;    -   repeating the Clifford stabilizer g_(A) measurement for the |ψ₁        state a threshold number of times;    -   preparing a Toffoli magic state in response to determining the        Clifford stabilizer g_(A) measurements are trivial; and    -   applying a sequence of Clifford gates to a logical input state        |ω        _(L) and the prepared Toffoli magic state to simulate the        Toffoli gate, wherein Clifford error corrections are applied to        the outputs of the sequence of Clifford gates applied to the        logical inputs.-   Clause 48. The system of clause 47, wherein the one or more    computing devices are further configured to implement:    -   growing the Toffoli magic state from a first code distance to a        second code distance, wherein the STOP algorithm is used to        measure stabilizers and minimum weight perfect matching (MWPM)        is applied to a measured syndrome history generated from        measuring the stabilizers to correct for errors.-   Clause 49. A method of preparing a Toffoli gate for use in quantum    computing, the method comprising:    -   preparing a plurality of Toffoli magic states, wherein        computational basis states used in preparing the Toffoli magic        states are encoded using a repetition code;    -   distilling the Toffoli gate from two or more of the prepared        Toffoli magic states, wherein distilling the Toffoli gate        comprises preparing a check qubit associated with the Toffoli        gate, wherein the check qubit indicates whether an error is        present in the distilled Toffoli gate; and    -   in response to verifying the check qubit does not indicate an        error, utilizing the distilled Toffoli gate to perform a logical        Toffoli gate operation.-   Clause 50. The method of clause 49, wherein distilling the Toffoli    gate from the two or more of the prepared Toffoli magic states,    comprises:    -   performing a plurality of rounds of lattice surgery operations        between qubits of a selected set of the plurality of Toffoli        magic states and qubits of the distilled Toffoli gate; and    -   wherein each of the rounds of lattice surgery acts on at least        one of the check qubits associated with the distilled Toffoli        gate.-   Clause 51. The method of clause 49, wherein the distilled Toffoli    gate has a fault rate of less than 1×10⁻⁶.-   Clause 52. The method of clause 49, wherein the distilled Toffoli    gate is distilled using 8 of the Toffoli magic states.-   Clause 53. The method of clause 52, wherein the two distilled    Toffoli gate have a probability of error that is less than a highest    probability of error of the respective ones of the 8 Toffoli magic    states reduced by a power of two.-   Clause 54. The method of clause 49, wherein the distilled Toffoli    gate is distilled using 2 of the Toffoli magic states.-   Clause 55. The method of clause 54, wherein the distilled Toffoli    gate has an error probability that is reduced by a power of two as    compared to respective error rates of the 2 Toffoli magic states,    when the 2 Toffoli magic states have highly biased noise.-   Clause 56. The method of clause 49, wherein the distilled Toffoli    gate is distilled using 8 of the Toffoli magic states, and    -   wherein the distilled Toffoli gate has an error probability that        is reduced by a power of three as compared to error rates of        respective ones of the 8 Toffoli magic states, when the 8        Toffoli magic states have highly biased noise.-   Clause 57. The method of clause 49, wherein a single round of    distillation is performed to distill the Toffoli magic state, and    wherein the single round of distillation comprises performing a    plurality of lattice surgery operations.-   Clause 58. The method of clause 49, wherein the Toffoli magic states    and the distilled Toffoli gate are implemented using a system    comprising:    -   mechanical linear resonators; and    -   one or more control circuits coupled with the mechanical linear        resonators,    -   wherein the one or more control circuits are configured to        stabilize an arbitrary coherent state superposition (cat state)        of the mechanical resonators to store quantum information of the        Toffoli magic states and the distilled Toffoli gate, wherein to        stabilize the arbitrary cat-state, the one or more control        circuits are configured to:        -   excite phonons in the mechanical resonators by driving            respective storage modes of the mechanical resonators; and        -   dissipate phonons from the mechanical resonators via one or            more respective open transmission lines of the one or more            control circuits coupled to the mechanical resonators,            wherein the open transmission line is configured to absorb            photons from the respective one or more control circuits.-   Clause 59. A system comprising:    -   mechanical resonators; and    -   one or more control circuits coupled with the mechanical        resonators, wherein the one or more control circuits are        configured to stabilize arbitrary coherent state superpositions        (cat states) of the mechanical resonators to store quantum        information; and    -   one or more computing devices storing program instructions, that        when executed cause the one or more control circuits to perform:        -   preparing a plurality of Toffoli magic states;        -   distilling a Toffoli gate from two or more of the prepared            Toffoli magic states, wherein distilling the Toffoli gate            comprises preparing a check qubit associated with the            Toffoli gate, wherein the check qubit indicates whether an            error is present in the distilled Toffoli gate; and        -   in response to verifying the check qubit does not indicate            an error, utilizing the distilled Toffoli gate to perform a            logical Toffoli gate operation.-   Clause 60. The system of clause 59, wherein the distilled Toffoli    gates comprise two distilled Toffoli gates that are distilled using    8 of the Toffoli magic states.-   Clause 61. The system of clause 60, wherein the two distilled    Toffoli gate have a probability of error that is less than a highest    probability of error of the respective ones of the 8 Toffoli magic    states reduced by a power of two.-   Clause 62. The system of clause 59, wherein the distilled Toffoli    gate is distilled using 2 of the Toffoli magic states.-   Clause 63. The system of clause 62, wherein the distilled Toffoli    gate has an error probability that is reduced by a power of two as    compared to respective error rates of the 2 Toffoli magic states,    when the 2 Toffoli magic states have highly biased noise.-   Clause 64. The system of clause 59, wherein the distilled Toffoli    gate is distilled using 8 of the Toffoli magic states, and    -   wherein the distilled Toffoli gate has an error probability that        is reduced by a power of three as compared to error rates of        respective ones of the 8 Toffoli magic states, when the 8        Toffoli magic states have highly biased noise.-   Clause 65. The system of clause 59, wherein the plurality of Toffoli    magic states uses as inputs are stabilized using a STOP algorithm    wherein to apply the STOP algorithm, the one or more computing    devices are configured to implement:    -   tracking consecutive syndrome outcomes;    -   computing a minimum number of faults capable of causing a        tracked sequence of consecutive syndrome outcomes;    -   stopping the STOP algorithm if either of the following        conditions is met:        -   1) a same syndrome outcome is repeated a threshold number of            times in a row, wherein the threshold is equal to one plus a            difference between:            -   a code distance of one of the fault-tolerant                computational basis states minus one wherein the result                of the subtraction is divided by two; and            -   a currently computed minimum number of faults capable of                causing the tracked sequence of consecutive syndrome                outcomes; or        -   2) the currently computed minimum number of faults capable            of causing the tracked sequence of consecutive syndromes is            equal to the code distance of the one of the fault-tolerant            computational basis states minus one wherein the result of            the subtraction is divided by two, and wherein one            additional round of syndrome measurements is performed            subsequently; and    -   utilizing the repeated syndrome if condition 1 is met or        utilizing the syndrome for the subsequently performed syndrome        measurement if condition 2 is met, wherein the utilized syndrome        it utilized to error correct the one of the fault-tolerant        computational basis states.-   Clause 66. The system of clause 59, wherein respective ones of the    one or more control circuits comprise:    -   an asymmetrically-threaded superconducting quantum interference        device (ATS) coupled with respective ones of the mechanical        resonators.-   Clause 67. A method of distilling a logical Toffoli gate from a    plurality of Toffoli magic states, the method comprising:    -   performing a plurality of rounds of lattice surgery operations        between qubits of a selected set of the plurality of Toffoli        magic states and qubits for a distilled Toffoli gate; and    -   wherein each of the rounds of lattice surgery acts on at least        one of the check qubits associated with the distilled Toffoli        gate.-   Clause 68. The method of clause 67, wherein    -   the distilled Toffoli gate is distilled using 8 of the Toffoli        magic states; and    -   the distilled Toffoli gate has a probability of error that is        less than a highest probability of error of the respective ones        of the 8 Toffoli magic states reduced by a power of two.-   Clause 69. A method of measuring an ancilla qubit in a context of    error correction of stored quantum information, wherein a set of one    or more error correction gates are applied between one or more data    qubits storing the quantum information and the ancilla qubit to    entangle the ancilla qubit with the one or more data qubits, the    method comprising:    -   transferring an excitation of the ancilla qubit to an additional        readout qubit using a SWAP gate or other sequence of one or more        gates that perform a swap function;    -   performing one or more measurements of the readout qubit; and    -   applying another set of one or more error correction gates        between the one or more data qubits storing the quantum        information and the ancilla qubit concurrently with performing        at least some of the one or more measurements of the readout        qubit.-   Clause 70. The method of clause 69, wherein the data qubits, the    ancilla qubit, and the readout qubit are implemented using    mechanical resonators.-   Clause 71. The method of clause 70, wherein the swap gate is    mediated by an asymmetrically threaded superconducting quantum    interference device (ATS).-   Clause 72. The method of clause 69, wherein the data qubits, the    ancilla qubit, and the readout qubit are implemented using bosonic    modes.-   Clause 73. The method of clause 69, wherein an amount of time during    which the one or more data qubits idle while performing the swap    gate is less than an amount of time required to perform the one or    more measurements of the readout qubit.-   Clause 74. The method of clause 73, wherein the one or more    measurements of the readout qubit:    -   comprise a plurality of repeated measurements taken subsequent        to performing the swap gate or other gates that perform the swap        function; and    -   are repeated up until an approximate time when a swap gate        operation is performed for a next round of error correction,        wherein the swap gate operation of the next round of error        correction is performed subsequent to applying the other set of        one or more error correction gates.-   Clause 75. The method of clause 74, wherein the plurality of    repeated measurements of the readout qubit comprise repeated QND    (quantum non demolition) parity measurements of the readout qubit.-   Clause 76. The method of clause 73, wherein the readout qubit is a    higher mode of an ancilla oscillator for the ancilla qubit.-   Clause 77. The method of clause 76, wherein the ancilla oscillator    is a λ/2 oscillator, and wherein the readout qubit has a mode that    is twice a base mode of the ancilla oscillator.-   Clause 78. A method of measuring a bosonic qubit wherein a    measurement outcome is affected by a single photon loss event, the    method comprising:    -   deflating the bosonic qubit, prior to performing a readout of        the bosonic qubit, such that phonons or photons are dissipated        from the bosonic qubit while a measurement observable of the        bosonic qubit is preserved; and    -   performing, subsequent to the deflating, a readout of the        measurement observable of the deflated bosonic qubit.-   Clause 79. The method of clause 78, wherein deflating the bosonic    qubit comprises:    -   changing a dissipater parameter such that an average photon        number or average phonon number of the bosonic qubit (α) is        reduced from an α_(initial) value to an α_(final) value, wherein        |α_(final)|<|α_(initial)|.-   Clause 80. The method of clause 78, wherein:    -   deflating the bosonic qubit comprises varying a steady state of        a two-photon dissipation process for the bosonic qubit; and    -   performing the readout of the measurement observable of the        deflated bosonic qubit comprises performing a parity readout of        the deflated bosonic qubit.-   Clause 81. The method of clause 78, wherein the bosonic qubit is    implemented using a system comprising:    -   mechanical resonators; and    -   a control circuit coupled with the mechanical resonators,    -   wherein the control circuit is configured to stabilize an        arbitrary coherent state superposition (cat state) of the        mechanical resonators to store quantum information, wherein to        stabilize the arbitrary cat-state, the control circuit is        configured to:        -   excite phonons in the mechanical resonators by driving            respective storage modes of the mechanical resonators; and        -   dissipate phonons via an open transmission line coupled to            the control circuit configured to absorb photons from a dump            mode of the control circuit.-   Clause 82. The method of clause 81, wherein the control circuit    comprises:    -   an asymmetrically-threaded superconducting quantum interference        device (ATS) coupled with the mechanical resonators, and    -   wherein deflating the bosonic qubit comprises changing a steady        state of a two photon dissipation controlled by the ATS.-   Clause 83. A method of performing a measurement of a first mode (a)    representing quantum information stored in a cat qubit, the method    comprising:    -   deflating the cat qubit such that an even number of phonons or        photons are dissipated from the cat qubit;    -   evolving the cat qubit under a Hamiltonian that couples a number        of excitations of the cat qubit to a change in a measurable        property of another mode (b); and    -   measuring the other mode (b).-   Clause 84. The method of clause 83, wherein:    -   the measurement is a determination of a parity of the cat qubit;    -   deflating the cat qubit comprises deflating the cat qubit such        that an average photon number or average phonon number of the        cat qubit (a) is reduced to zero, wherein an even cat state is        mapped to |0        and an odd cat state is mapped to |1        ;    -   the Hamiltonian is selected from a three or higher wave mixing        Hamiltonian that correlates phonon number or photon number to a        change of the other mode (b); and    -   measuring the other mode (b) using homodyne, heterodyne, or        photo detection.-   Clause 85. The method of clause 83, wherein the Hamiltonian selected    from the three or higher wave mixing Hamiltonian comprises    ig(b^(†)−b)a^(†)a.-   Clause 86. The method of clause 83, wherein the Hamiltonian selected    from the three or higher wave mixing Hamiltonian comprises    g(b^(†)+b)a^(†)a.-   Clause 87. The method of clause 83, wherein the Hamiltonian selected    from the three or higher wave mixing Hamiltonian comprises a product    of a^(†)a with a term that affects the other mode (b) in a    measureable way.-   Clause 88. The method of clause 83, wherein:    -   the cat qubit is implemented via a mechanical resonator;    -   the other mode (b) is a dump mode; and    -   the Hamiltonian is selected from a three or higher wave mixing        Hamiltonian that correlates the average phonon number or the        average photon number to a change of the other mode (b), wherein        the three wave mixing is mediated by an ATS.-   Clause 89. A method of performing a measurement of quantum    information in a cat qubit, the method comprising:    -   evolving under a Hamiltonian which couples the phase of a of the        cat qubit (an “a” mode) to a measurable property of another        bosonic mode (a “b” mode) wherein the Hamiltonian is achieved        via a three wave or higher mixing Hamiltonian; and    -   performing homodyne, heterodyne, or photo detection of the “b”        mode to determine a state of the “a” mode,-   wherein the cat qubit is implemented using a system comprising:    -   mechanical resonators; and    -   a control circuit comprising an asymmetrically-threaded        superconducting quantum interference device (ATS) coupled with        the mechanical resonators,    -   wherein the control circuit is configured to stabilize an        arbitrary coherent state superposition (cat state) of the        mechanical resonators to store quantum information, wherein to        stabilize the arbitrary cat-state, the control circuit is        configured to:        -   excite phonons in the mechanical resonators by driving            respective storage modes of the mechanical resonators; and        -   dissipate phonons via an open transmission line coupled to            the control circuit configured to absorb photons from a dump            mode of the control circuit.-   Clause 90. The method of clause 89, wherein:    -   the Hamiltonian is derived from a three wave mixing Hamiltonian        mediated by an ATS;    -   the “a” mode is implemented via a mechanical storage resonator;        and    -   the “b” mode is implemented via an electromagnetic resonator.-   Clause 91. The method of clause 90, wherein a Hamiltonian for the    readout comprises:    g(a+a ^(†))(b+b ^(†));    ig(a+a ^(†))(b−b ^(†)); or    g(ab ^(†) +a ^(†) b).    Illustrative Computer System

FIG. 39 is a block diagram illustrating an example computing device thatmay be used in at least some embodiments.

FIG. 39 illustrates such a general-purpose computing device 3900 as maybe used in any of the embodiments described herein. In the illustratedembodiment, computing device 3900 includes one or more processors 3910coupled to a system memory 3920 (which may comprise both non-volatileand volatile memory modules) via an input/output (I/O) interface 3930.Computing device 3900 further includes a network interface 3940 coupledto I/O interface 3930.

In various embodiments, computing device 3900 may be a uniprocessorsystem including one processor 3910, or a multiprocessor systemincluding several processors 3910 (e.g., two, four, eight, or anothersuitable number). Processors 3910 may be any suitable processors capableof executing instructions. For example, in various embodiments,processors 3910 may be general-purpose or embedded processorsimplementing any of a variety of instruction set architectures (ISAs),such as the ×86, PowerPC, SPARC, or MIPS ISAs, or any other suitableISA. In multiprocessor systems, each of processors 3910 may commonly,but not necessarily, implement the same ISA. In some implementations,graphics processing units (GPUs) may be used instead of, or in additionto, conventional processors.

System memory 3920 may be configured to store instructions and dataaccessible by processor(s) 3910. In at least some embodiments, thesystem memory 3920 may comprise both volatile and non-volatile portions;in other embodiments, only volatile memory may be used. In variousembodiments, the volatile portion of system memory 3920 may beimplemented using any suitable memory technology, such as static randomaccess memory (SRAM), synchronous dynamic RAM or any other type ofmemory. For the non-volatile portion of system memory (which maycomprise one or more NVDIMMs, for example), in some embodimentsflash-based memory devices, including NAND-flash devices, may be used.In at least some embodiments, the non-volatile portion of the systemmemory may include a power source, such as a supercapacitor or otherpower storage device (e.g., a battery). In various embodiments,memristor based resistive random access memory (ReRAM),three-dimensional NAND technologies, Ferroelectric RAM, magnetoresistiveRAM (MRAM), or any of various types of phase change memory (PCM) may beused at least for the non-volatile portion of system memory. In theillustrated embodiment, program instructions and data implementing oneor more desired functions, such as those methods, techniques, and datadescribed above, are shown stored within system memory 3920 as code 3925and data 3926.

In some embodiments, I/O interface 3930 may be configured to coordinateI/O traffic between processor 3910, system memory 3920, and anyperipheral devices in the device, including network interface 3940 orother peripheral interfaces such as various types of persistent and/orvolatile storage devices. In some embodiments, I/O interface 3930 mayperform any necessary protocol, timing or other data transformations toconvert data signals from one component (e.g., system memory 3920) intoa format suitable for use by another component (e.g., processor 3910).In some embodiments, I/O interface 3930 may include support for devicesattached through various types of peripheral buses, such as a variant ofthe Peripheral Component Interconnect (PCI) bus standard or theUniversal Serial Bus (USB) standard, for example. In some embodiments,the function of I/O interface 3930 may be split into two or moreseparate components, such as a north bridge and a south bridge, forexample. Also, in some embodiments some or all of the functionality ofI/O interface 3930, such as an interface to system memory 3920, may beincorporated directly into processor 3910.

Network interface 3940 may be configured to allow data to be exchangedbetween computing device 3900 and other devices 3960 attached to anetwork or networks 3950, such as other computer systems or devices. Invarious embodiments, network interface 3940 may support communicationvia any suitable wired or wireless general data networks, such as typesof Ethernet network, for example. Additionally, network interface 3940may support communication via telecommunications/telephony networks suchas analog voice networks or digital fiber communications networks, viastorage area networks such as Fibre Channel SANs, or via any othersuitable type of network and/or protocol.

In some embodiments, system memory 3920 may represent one embodiment ofa computer-accessible medium configured to store at least a subset ofprogram instructions and data used for implementing the methods andapparatus discussed in the context of FIG. 1 through FIG. 38. However,in other embodiments, program instructions and/or data may be received,sent or stored upon different types of computer-accessible media.Generally speaking, a computer-accessible medium may includenon-transitory storage media or memory media such as magnetic or opticalmedia, e.g., disk or DVD/CD coupled to computing device 3900 via I/Ointerface 3930. A non-transitory computer-accessible storage medium mayalso include any volatile or non-volatile media such as RAM (e.g. SDRAM,DDR SDRAM, RDRAM, SRAM, etc.), ROM, etc., that may be included in someembodiments of computing device 3900 as system memory 3920 or anothertype of memory. In some embodiments, a plurality of non-transitorycomputer-readable storage media may collectively store programinstructions that when executed on or across one or more processorsimplement at least a subset of the methods and techniques describedabove. A computer-accessible medium may further include transmissionmedia or signals such as electrical, electromagnetic, or digitalsignals, conveyed via a communication medium such as a network and/or awireless link, such as may be implemented via network interface 3940.Portions or all of multiple computing devices such as that illustratedin FIG. 39 may be used to implement the described functionality invarious embodiments; for example, software components running on avariety of different devices and servers may collaborate to provide thefunctionality. In some embodiments, portions of the describedfunctionality may be implemented using storage devices, network devices,or special-purpose computer systems, in addition to or instead of beingimplemented using general-purpose computer systems. The term “computingdevice”, as used herein, refers to at least all these types of devices,and is not limited to these types of devices.

CONCLUSION

Various embodiments may further include receiving, sending or storinginstructions and/or data implemented in accordance with the foregoingdescription upon a computer-accessible medium. Generally speaking, acomputer-accessible medium may include storage media or memory mediasuch as magnetic or optical media, e.g., disk or DVD/CD-ROM, volatile ornon-volatile media such as RAM (e.g. SDRAM, DDR, RDRAM, SRAM, etc.),ROM, etc., as well as transmission media or signals such as electrical,electromagnetic, or digital signals, conveyed via a communication mediumsuch as network and/or a wireless link.

The various methods as illustrated in the Figures and described hereinrepresent exemplary embodiments of methods. The methods may beimplemented in software, hardware, or a combination thereof. The orderof method may be changed, and various elements may be added, reordered,combined, omitted, modified, etc.

Various modifications and changes may be made as would be obvious to aperson skilled in the art having the benefit of this disclosure. It isintended to embrace all such modifications and changes and, accordingly,the above description to be regarded in an illustrative rather than arestrictive sense.

What is claimed is:
 1. A method comprising: simulating a cat qubit,wherein the simulating comprises: defining basis states for the catqubit to be simulated; orthonormalizing the defined basis states toconstruct 2d orthonormalized shifted Fock basis states for the cat qubitto be simulated; determining matrix elements of an operator in theorthonormalized shifted Fock basis states; constructing an annihilationoperator for the cat qubit to be simulated in the orthonormalizedshifted Fock basis using the determined matrix elements of the operator;implementing a physical cat qubit based on the simulated cat qubit,wherein: Z error rates for gates associated with the physical cat qubitare determined based at least in part on the annihilation operator; andone or more characteristics of the physical cat qubit or the gates aredetermined based at least in part on the Z error rates.
 2. The method ofclaim 1, wherein the defined basis states, before performing theorthonormalization, are defined such that the defined basis states aregrouped into even and odd branches.
 3. The method of claim 2, wherein,in a ground state, normalized versions of the defined basis states areequivalent to complementary basis states of the cat qubit to besimulated expressed as 1+

or 1−

instead of computational basis states expressed as |0

or |1

.
 4. The method of claim 3, wherein the defined basis states indifferent parity sectors are orthogonal to one another such that theorthonormalization is performed separately in the respective paritysectors.
 5. The method of claim 1, wherein simulating the cat qubitfurther comprises: simulating Pauli Z errors for the cat qubitsimulation in the 2 d orthonormalized shifted Fock basis states.
 6. Themethod of claim 1, wherein the physical cat qubit is a hybridacoustic-electrical qubit implemented using a linear mechanicalresonator.
 7. The method of claim 1, wherein the physical cat qubit isimplemented using an electromagnetic resonator.
 8. One or morenon-transitory computer-readable media storing program instructions,that when executed on or across one or more processors, cause the one ormore processors to: simulate a cat qubit, wherein the simulationcomprises: define basis states for a cat qubit to be simulated;orthonormalize the defined basis states to construct 2 d orthonormalizedshifted Fock basis states for the cat qubit to be simulated; determinematrix elements of an operator in the orthonormalized shifted Fock basisstates; and construct an annihilation operator for the cat qubit to besimulated in the orthonormalized shifted Fock basis using the determinedmatrix elements of the operator; cause a physical cat qubit to beimplemented based on the simulated cat qubit, wherein: Z error rates forgates associated with the physical cat qubit are determined based atleast in part on the annihilation operator; and one or morecharacteristics of the physical cat qubit or the gates are determinedbased at least in part on the Z error rates.
 9. The one or morenon-transitory computer-readable media of claim 8, wherein the programinstructions, when executed on or across the one or more processors,further cause the one or more processors to: simulate Pauli Z errors forthe cat qubit simulation in the 2 d orthonormalized shifted Fock basisstates.
 10. The one or more non-transitory computer-readable media ofclaim 8, wherein the defined basis states, before performing theorthonormalization, are defined such that the defined basis states aregrouped into even and odd branches.
 11. The one or more non-transitorycomputer-readable media of claim 8, wherein, in a ground state,normalized versions of the defined basis states are equivalent tocomplementary basis states of the cat qubit to be simulated expressed as|+

or |−

instead of computational basis states expressed as |0

or |1

.
 12. The one or more non-transitory computer-readable media of claim 8,wherein the defined basis states are orthogonal to one another such thatthe orthonormalization is performed separately in respective paritysectors.
 13. A system, comprising: a memory storing programinstructions; and one or more processors, wherein the programinstructions, when executed on or across the one or more processorscause the one or more processors to: simulate a cat qubit, wherein thesimulation comprises: define basis states for a cat qubit to besimulated; orthonormalize the defined basis states to construct 2 dorthonormalized shifted Fock basis states for the cat qubit to besimulated; determine matrix elements of an operator in theorthonormalized shifted Fock basis states; and construct an annihilationoperator for the cat qubit to be simulated in the orthonormalizedshifted Fock basis using the determined matrix elements of the operator;cause a physical cat qubit to be implemented based on the simulated catqubit, wherein: Z error rates for gates associated with the physical catqubit are determined based at least in part on the annihilationoperator: and one or more characteristics of the physical cat qubit orthe gates are determined based at least in part on the Z error rates.14. The system of claim 13, wherein the program instructions, whenexecuted on or across the one or more processors, further cause the oneor more processors to: simulate Pauli Z errors for the cat qubitsimulation in the 2 d orthonormalized shifted Fock basis states.
 15. Thesystem of claim 13, wherein the defined basis states, before performingthe orthonormalization, are defined such that the defined basis statesare grouped into even and odd branches.
 16. The system of claim 13,wherein, in a ground state, normalized versions of the defined basisstates are equivalent to complementary basis states of the cat qubit tobe simulated expressed as |+

or |−

instead of computational basis states expressed as |0

or |1

.
 17. The system of claim 13, wherein the defined basis states areorthogonal to one another such that the orthonormalization is performedseparately in respective parity sectors.
 18. The system of claim 13,wherein the physical cat qubit is implemented using mechanicalresonators.
 19. The system of claim 13, wherein the physical cat qubitis implemented using electromagnetic resonators.
 20. The system of claim13, wherein the physical cat qubit is implemented in a system comprisingone or more mechanical resonators and one or more electromagneticresonators.